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2024 | Book

# Vehicle Vibrations

## Linear and Nonlinear Analysis, Optimization, and Design

Authors: Reza N. Jazar, Hormoz Marzbani

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​Vehicle Vibrations: Linear and Nonlinear Analysis, Optimization, and Design is a self-contained textbook that offers complete coverage of vehicle vibration topics from basic to advanced levels. Written and designed to be used for automotive and mechanical engineering courses related to vehicles, the text provides students, automotive engineers, and research scientists with a solid understanding of the principles and application of vehicle vibrations from an applied viewpoint. Coverage includes everything you need to know to analyze and optimize a vehicle’s vibration, including vehicle vibration components, vehicle vibration analysis, flat ride vibration, tire-road separations, and smart suspensions.

#### Part I

##### Chapter 1. Springs
Abstract
A spring is any elastic body whose function is to collect potential energy by elastic deformation and recover its original shape and dimensions after the removal of the load. The primary purpose of springs in vehicle suspensions is to extend the period over which energy is absorbed and subsequently released. In vehicle applications, the most commonly used springs for suspension systems are helical springs, leaf springs, and torsion bars. Their deformation under loads are either in bending or twisting, and hence, the stored potential strain energy will be made by normal stress in bending or shear stress in torsion. Figure 1.1 illustrates a coil spring.
Reza N. Jazar, Hormoz Marzbani
##### Chapter 2. Dampers
Abstract
A liquid shock absorber is typically an oil pump placed between the frame of the vehicle and the wheels. The upper mount of the shock connects to the frame, i.e., the sprung mass, while the lower mount connects to the axle near the wheel, i.e., the unsprung mass as is shown in Fig. 2.1.
Reza N. Jazar, Hormoz Marzbani
##### Chapter 3. Suspension Systems
Abstract
Every multi-wheel vehicle requires a suspension system to provide road contact to all wheels and provide ride comfort. Due to physical and engineering imperfections, it is impossible that all wheels of a multi-wheel rigid vehicle touch the ground at the same time. Flexible suspension is required to make all wheels to be in contact with the ground on road irregularities, wheels’ unequal loads, and under forward and lateral accelerating conditions. Figure 3.1 illustrates side view of a solid axle suspension with leaf spring. The mechanical requirements of suspensions and their design will be studied in this chapter.
Reza N. Jazar, Hormoz Marzbani
##### Chapter 4. Mechanical Vibrations Modeling
Abstract
In this chapter, we study 1—mechanical elements of vibrating systems, 2—physical causes of mechanical vibrations, 3—kinematics of vibrations, 4—simplification methods of complex vibrating systems, and 5—mathematical review of mechanical vibrations. We will solve and review a general mass–spring–damper system and then will make other vibrating systems to be reduced to an equivalent system. Such equivalence yields a generalization in mathematical treatment of vibrating systems. The first section defines all fundamental terms we need to study vehicle vibrations.
Reza N. Jazar, Hormoz Marzbani

#### Part II

##### Chapter 5. Vibration Dynamics
Abstract
In this chapter, we review the dynamics of vibrations and the methods of deriving the equations of motion of vibrating systems. The Newton–Euler and Lagrange methods are the most applied methods of deriving the equations of motion. Having symmetric coefficient matrices for multi-degrees-of-freedom linear vibrating systems is the main advantage of using the Lagrange method in mechanical vibrations. For example, Fig. 5.1 illustrates a one degree of freedom vibrating system.
Reza N. Jazar, Hormoz Marzbani
##### Chapter 6. Time Response

The time response of a vibrating system refers to its reaction when subjected to nonzero initial conditions and/or time-varying forcing functions, known as transient excitations. This behavior is referred to as transient response because it captures the system’s response during a finite period of time.

Reza N. Jazar, Hormoz Marzbani
##### Chapter 7. Frequency Response
Abstract
The harmonic excitation is the most widely used continuous excitation in vehicle vibration applications. Any combination of $$\sin$$ and $$\cos$$ functions is called a harmonic function. Frequency response analysis is simple and straightforward because the response of linear discrete vibrating systems to harmonic excitation is harmonic with proportional magnitude. The magnitudes of harmonic responses are functions of the excitation frequencies and depend on the system parameters. The goal of frequency response analysis is to find and plot the magnitude of the harmonic response versus excitation frequency. Such a function is called the frequency response and indicates the behavior of the system at different excitation frequencies. Figure 7.1 illustrates the only four types of harmonically excited one-DOF systems.
Reza N. Jazar, Hormoz Marzbani

#### Part III

##### Chapter 8. 1/8 Vehicle Model
Abstract
The simplest vibrating model of vehicles is considering only vertical vibration of the vehicle while ignoring the flexibility of tires compared to the stiffness of the main springs. Let us cut a vehicle longitudinally and laterally, and we keep one quarter of the vehicle, and then, assume that tires are rigid and massless. This way we model a vehicle with a one degree of freedom (DOF) base excited system with the mass m to be one quarter of the mass of the body of vehicle, and suspension spring k and damper c to be equal to k and c of one side of the vehicle.
Reza N. Jazar, Hormoz Marzbani
##### Chapter 9. Quarter Car Model and Body Bounce Mode
Abstract
The most useful and practical vibration model of a vehicle suspension system is the quarter car model, shown in Fig. 9.1. Quarter car model is excellent to examine and optimize the body bounce mode of vibrations. This model includes the body bounce $$x_{s}$$, wheel hop $$x_{u}$$, and the road excitations y. We introduce, examine, and optimize the quarter car model in this section.
Reza N. Jazar, Hormoz Marzbani
##### Chapter 10. Bicycle Car Vibration Model and Body Pitch Mode
Abstract
The simplest vehicle model to study the pitch mode of vibrations is the bicycle model in which we longitudinally cut a vehicle and take one side to study. Figure 10.1 illustrates a bicycle vibrating model of a vehicle. This model includes the body bounce x, body pitch $$\theta$$, wheels hop $$x_{1}$$ and $$x_{2}$$, and the road excitations $$y_{1}$$ and $$y_{2}$$.
Reza N. Jazar, Hormoz Marzbani
##### Chapter 11. Half Car Model and Body Roll Mode
Abstract
To examine and optimize the roll vibrations of a vehicle, we use the half car vibrating model in which we laterally cut a vehicle and take one half to study. Figure 11.1 illustrates a half car model. This model includes the body bounce x, body roll $$\varphi$$, and wheels hop $$x_{1}$$ and $$x_{2}$$. There are also two independent road excitations $$y_{1}$$ and $$y_{2}$$ under the left and right wheels. The roll vibration of a car is the most unpleasant and the most uncomfortable motion of a vehicle.
Reza N. Jazar, Hormoz Marzbani
##### Chapter 12. Full Car Vibrating Model
Abstract
The full car model, also known as a general vibrating model of a vehicle, encompasses the complete representation of a vehicle’s vibrations, including the bounce, pitch, and roll motions of the body, as well as the vibrations of the wheels. In the case of a four-wheeled vehicle, the full car model incorporates seven degrees of freedom: three for the vehicle body and four for the individual wheels. Figure 12.1 provides a visual depiction of a full car model with four wheels.
Reza N. Jazar, Hormoz Marzbani

#### Part IV

##### Chapter 13. Flat Ride Tuning
Abstract
Flat ride refers to a phenomenon where the uncomfortable pitching oscillation of a vehicle transforms into a more tolerable bouncing oscillation when the vehicle encounters a bump while moving forward.
Reza N. Jazar, Hormoz Marzbani
Abstract
Tire–road separation is an infrequent and extraordinary occurrence that arises during the analysis of a vehicle’s ride and handling. It typically manifests as a transient phenomenon that quickly resolves itself.
Reza N. Jazar, Hormoz Marzbani
##### Backmatter
Title
Vehicle Vibrations
Authors
Reza N. Jazar
Hormoz Marzbani