1 Introduction

Ride vibration is a principal concern for vehicle dynamics, where frame flexibility plays a very significant role. Numerous studies have been conducted on problems of ride vibration. However, very few studies have been carried out to study the effect of vibration, which includes a flexible frame. It has been generally noticed that the flexural vibration can be neglected for smaller vehicles, relatively for stiffer automobiles. However, trucks and the long heavy vehicle experience significant beaming mode vibration (Margolis and Edeal 1989). Ibrahim (1996) has created a model of truck frame through FEM where he calculated the modal properties of the frame. The results were presented in terms of PSD and RMS of the vehicle vertical response. Most of the analyses on the effects of chassis flexibility in vehicle dynamics were based on the use of particular model developed, whereas the FEM (finite element method) and the modal superposition theory have been extensively used to calculate the modal properties of the frame structure. Generally, the problem occurs due to the beam flexibility of the vehicle, which is termed as ‘beaming response of the vehicle’ (Rideout 2012). Rideout and Khan (2010) presented the model of the frame flexibility issue for truck model. They created a pitch plane model which involved longitudinal dynamics and transverse frame vibrations. Huang and Zeng (2010) studied about flexural vibration problems for high-speed passenger rail vehicle, where car body was modeled through Euler beam approach. This research underlined the effect of internal damping in ride comfort of rail vehicle. Kumar et al. (2017) analyzed the whole-body vibration for railway system. They have considered biodynamic model of human for the assessment of ride comfort, where whole-body model was integrated with flexible bogie model system. The flexible bogie system was modeled through Euler–Bernoulli beam approach. Similarly in other studies, Euler–Bernoulli approach was attempted to introduce beam flexibility in heavy truck chassis modeling (Zhou et al. 2009; Kim and Yim 1994). Heavy long vehicles have a ladder-type chassis structure across the whole length of the vehicle. These structures are stiffer in bending, which can respond to bending frequencies more than 20 Hz. When the engine and other elements are mounted on the structure, these bending frequencies are reduced up to 8 Hz, for highly stiffer frames (Cai-hong and Zeng 2010; Tomioka et al. 2017). Tomioko and Takigami (2015) have conducted the experimental study on flexural beam to evaluate the effect of frame flexibility on passenger comfort. Some other studies apparently show that the beaming frequency is higher than the rigid frequency of the vehicle (Tomioka et al. 2006; Agostinacchio et al. 2014). However, in large vehicles, beaming frequency is influenced by road inputs.

A lot of research has been conducted considering the issue of frame flexibility in the analysis of railway vehicle. But, limited studies were reported on dynamic analysis of heavy road vehicles considering random road disturbances. A country like India, where multifarious road conditions are present, which creates an infirm condition before all leading automobile industries to maintain comfort level as per standard on every road conditions. Some studies have mentioned that structural damping can participate in ride quality improvement. So, in the present work frame flexibility has been incorporated by considering frame as Raleigh beam, which also includes the effect of the rotary inertia. Lagrangian approach has been applied to obtain the energy equations for the vehicle system, which resulted from the well-known rigid body modes and the modal parameters of the frame flexibility. The work involves the importance of structural or internal damping of beam-type chassis structure on flexural vibration and ride quality of the heavy vehicle with random road disturbance prescribed by ISO 8608 (Meirovitch 1980) as input. The bond graph approach has been utilized to model the vehicle with flexible frame. Besides the beam response, the effect of different road conditions as per ISO 8608 and vehicle speeds on the dynamics of vehicle has also been presented. The work begins with deriving Lagrange equation of frame considered as Rayleigh beam for truck flexural response. Then, this formulation is being extended for bond graph modeling with or without consideration of structural damping effect. Simulations were carried out to visualize the beaming mode response with respect to random road conditions as prescribed by ISO 8608.

2 Analytical Framework for Heavy Vehicle System Through Classical Mechanics

2.1 Generalized Lagrange’s Equation for a Continuous System

It is a well-known fact that the Lagrange formulation provides one of the most convenient ways of writing down the equation of motion for a wide range of mechanical systems. The principal advantage of Lagrange’s equation is that it is easier to apply to dynamical systems other than the simplest. This formulation is created from scalar quantities of kinetic energy, potential energy and work expressed in terms of generalized coordinates and developed for universal handling of dynamical systems (Mukherjee and Karmakar 2000). The equation is formulated equations of motion from the principle of least action. This density function is a function comprised of time, transverse displacements and transverse velocities, and up to second derivatives of displacements to space coordinates x, which may be expressed as

$$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{L} \left( {w(t,x),\dot{w}(t,x),w^{\prime } (t,x),w^{\prime \prime } (t,x),\dot{w}^{\prime } (t,x)} \right) = \hat{T}(t,x) - \hat{V}(t,x)$$
(1)

where \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{L} ( \cdot )\) is the Lagrangian density function and \(\hat{T}( \cdot )\) is the kinetic energy density, which may be expressed as,

$$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{T} (x,t) = \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{T} (\dot{w}(x,t),\dot{w}^{{\prime }} (x,t))$$
(2)

Moreover, \(\hat{V}(t,x)\) is the potential energy density, which has the following functional dependence,

$$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{V} (x,t) = \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{V} (w^{\prime } (x,t),w^{\prime \prime } (x,t))$$
(3)

The next step is used for finding the equation of motion and the boundary conditions for the beam through the variational principle (see Appendix 2). This formulation yields

$$\delta \int\limits_{t1}^{t2} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{L} \left( {w(t,x),\dot{w}(t,x),w^{{\prime }} (t,x),w^{{\prime \prime }} (t,x),\dot{w}^{{\prime }} (t,x)} \right){\text{d}}t = 0}$$
(4)

where δ(·) is infinitesimal variation operator and it is very similar to the total derivative operator; the only difference is that δ(·) does not vary with time. Another significant property of infinitesimal operator is that it commutes with the differential operator. Using the extremization condition, one may obtain

$$\int_{0}^{L} {\int_{t1}^{t2} {\delta \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{L} \left( {w(t,x),\dot{w}(t,x),w^{{\prime }} (t,x),w^{{\prime \prime }} (t,x),\dot{w}^{{\prime }} (t,x)} \right)\,{\text{d}}x{\text{d}}t = 0} }$$
(5)

or

$$\int_{0}^{L} {\int_{t1}^{t2} {\left[ {\frac{{\partial \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{L} }}{\partial w}\delta w + \frac{{\partial \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{L} }}{{\partial \dot{w}}}\delta \dot{w} + \frac{{\partial \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{L} }}{{\partial w^{{\prime }} }}\delta w^{{\prime }} + \frac{{\partial \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{L} }}{{\partial \dot{w}^{\prime}}}\delta \dot{w}^{{\prime }} + \frac{{\partial \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{L} }}{{\partial w^{{\prime \prime }} }}\delta w^{{\prime \prime }} } \right]} } \,{\text{d}}x\,{\text{d}}t = 0$$
(6)

The detailed derivation of the variational formulation is presented in “Appendix 1.”

2.2 Analytical Formulation for Flexural Behavior of the Truck Chassis Structure

An analytical formulation for flexural behavior of the truck structure is being attempted in this section. The vehicle base is modeled as a rigid body with a local coordinate reference frame (w, x) attached to the center of mass (M) and aligned with the inertia principal axes as shown in Fig. 1. It has density (ρ) and pitch inertia (J) with respect to the body \(W_{1} (t) - W_{2} (t)\) displacement axis, EI flexural rigidity and θ(t) pitch velocity of the base. It is connected with the suspension system to different elements of the vehicles. The spring stiffness characterizes the spring and damper in the suspension system ks and damping coefficient Rs, respectively. The various displacements of the vehicle are described w(x, t) with respect to the equilibrium positions. As the vehicle is assumed to be rigid, its motion can be described by the vertical displacement (bounce) as w(x, t) and the rotation about the transverse horizontal axis (pitch) as \(\dot{\theta }(t)\).

Fig. 1
figure 1

Schematic diagram of a continuous frame of vehicle

Full size image

The model of the vehicle is being created with the following assumptions:

  • The components of the vehicle body act as a rigid body.

  • The springs and dampers of the suspension system have linear characteristics.

  • The spring damper system is assumed to be massless.

  • The tire is assumed to provide stiffness and thus modeled as a spring element.

  • The positions of the two ends of the spring connecting two rigid bodies or connecting one rigid body and one contact point are required as input data.

  • The spring damper system connects each rigid body.

  • The traveling road is assumed to be straight.

  • The road inputs are assumed to be random.

The analytical framework of truck chassis can be assumed as a free–free beam structure, where Rayleigh beam approach is being adopted. Let us consider a transverse deflection of the beam w(x,t), where x varies from 0 to beam length L. The formalism of beam flexibility is easily obtained through the well-known Lagrangian equation. Lagrangian density variation may be obtained from the following expression as

$$L = \left[ {\frac{1}{2}\int\limits_{0}^{L} {\left( {\rho A\dot{w}\,(x,t)^{2} + \rho I\dot{w}^{{\prime }} (x,t)^{2} } \right){\text{d}}x - \frac{1}{2}\int\limits_{0}^{L} {\left( {EIw^{{\prime \prime }} (x,t)^{2} } \right){\text{d}}x} } } \right]$$
(7)

where ρ is the mass density, A is the area of cross section, EI is the rigidity and I is the second moment of area of the cross section of the beam about the neutral axis. In Eq. (7), the first term represents kinetic energy of the beam, the second term presents as inertial energy and the final term is strain energy. The variational formulation presents a refined and classical method of deriving the equation of motion of a dynamical system, where all the boundary conditions of a system can be revealed. However, fundamentals of variational formulation for the continuous system are detailed in “Appendix 2.”

The equation of motion is obtained after applying a variational formulation and boundary conditions in free vibration condition, which may be expressed as

$$\rho A\ddot{w}(x,t) - \left( {\rho I\dot{w}^{{\prime \prime }} (x,t)} \right)^{{\prime }} + \left( {EIw^{{\prime \prime }} (x,t)} \right)^{{\prime \prime }} = 0$$
(8)

This equation of motion is based on Rayleigh beam model. The term \(\left( {EIw^{{\prime \prime }} (x,t)} \right)^{{\prime \prime }}\) is usually referred to as the flexural term, where EI is called the flexural stiffness and \(\left( {\rho I\dot{w}^{{\prime \prime }} (x,t)} \right)^{{\prime }}\) is known as the rotary inertia term. If this term is zero, then Eq. (8) will become a Euler beam model. Thus, the term \(\left( {\rho I\dot{w}^{{\prime \prime }} (x,t)} \right)^{{\prime }}\) can also be named as a Rayleigh beam term.

Assume a modal solution for Eq. (8) in the form of

$$w(x,t) = W(x)e^{i\omega t}$$
(9)

where ω is the eigenfrequency and W(x) is the eigenfunction. The actual solution of the equation may be obtained by taking the real part of the solution. Substituting the modal solution in the field Eq. (8) yields after rearrangement as

$$- \omega^{2} \left[ {\rho AW - \left( {\rho IW^{{\prime }} } \right)^{{\prime }} } \right] + \left( {EIW^{{\prime \prime }} } \right)^{{\prime \prime }} = 0$$
(10)

which along with the corresponding boundary condition represents the eigenvalue problem for a Rayleigh beam.

2.3 Modal Analysis for Uniform Rayleigh Beam

Considering Eq. (10) as a uniform beam and substituting in Eq. (9) a solution of the form

$$W(x) = Ce^{\lambda x}$$
(11)

where C and λ are constants, Eq. (10) can be rewritten as

$$EI\lambda^{4} - \omega^{2} \rho I\lambda^{2} - \omega^{2} \rho A = 0$$
(12)

yielding

$$\lambda^{2} = \frac{1}{2EI}\left[ {\omega^{2} \rho I \pm \sqrt {\omega^{4} \rho^{2} I^{2} + 4\omega^{2} EI\rho A} } \right]$$

Thus, the spatial equation (real form) of Eq. (12) has the general solution, which may be expressed as

$$W(x) = A\cosh \lambda x + B\sinh \lambda x + C\cos \lambda x + D\sin \lambda x$$
(13)

Here, one may consider a uniform Rayleigh beam for which the free–free boundary conditions are given by

$$W(0) = 0,\quad W^{{\prime }} (0) = 0,\quad W^{{\prime \prime }} (l) = 0,\quad W^{{{\prime \prime \prime }}} (l) = 0,$$
(14)

Using, Eq. (14) in Eq. (13) yields the frequency equation as

$$\cosh \lambda_{n} l\cos \lambda_{n} l - 1 = 0$$
(15)

The solution of Eq. (15) yields

$$\lambda_{n} = \left( {\frac{2n + 1}{2}\pi } \right)\frac{1}{l}$$

Moreover, the mode shape function may be expressed as

$$ \begin{aligned}W_{n} (x) &= (\cos \lambda_{n} l - \cosh \lambda_{n} l)(\sin \lambda_{n} x + \sinh \lambda_{n} x) \\ &\quad- (\sin \lambda_{n} l - \sinh \lambda_{n} l)(\cos \lambda_{n} x + \cosh \lambda_{n} x)\end{aligned} $$
(16)

Thus, one may obtain the natural frequency of Eq. (6), which may be written as

$$\omega_{n} = \lambda_{n}^{2} \frac{1}{{\left[ {1 + \lambda_{n}^{2} \frac{I}{A}} \right]^{1/2} }}\sqrt {\frac{EI}{\rho A}}$$
(17)

It is recognized that conditions are force-free at this point, which permits ωn = 0 to be a mode frequency. Thus, Eq. (10) will be written as

$$\frac{{{\text{d}}^{4} W}}{{{\text{d}}x^{4} }} = 0$$
(18)

or

$$W = c_{1} x^{3} + c_{2} x^{2} + c_{1} x + c_{4}$$
(19)

There are two possible solutions for Eq. (13), which satisfied all boundary conditions. They are W = const and W = ax + b. These are called rigid body modes, and it is convenient to assume them as rigid body rotation of the beam about the centrally located center of mass. Thus,

$$W_{00} (x) = 1$$
(20)

and

$$W_{0} (x) = x - \frac{L}{2}$$
(21)

Let us consider an analysis for truck chassis, which is presented in Fig. 1. In this figure, the half model of the road truck is supported by front and rear suspension, which exerted an external force on the beam over the various road responses. From equation of motion, Eq. (8) will be rewritten as

$$ \begin{aligned}&\rho A\ddot{w}(x,t) - \left( {\rho I\dot{w}^{{\prime \prime }} (x,t)} \right)^{{\prime }} + \left( {EIw^{{\prime \prime }} (x,t)} \right)^{{\prime \prime }} \\ &= F_{1} \delta (x - x_{1} ) + F_{2} \delta (x - x_{2} ) + F_{3} \delta (x - x_{3} )\end{aligned} $$
(22)

where F1 and F2 are the force exerted by front and rear suspension systems, respectively, and F3 is dead weight on the beam, δ(·) is the Dirac delta function, x1 and x2 are the respective distance from left end to the front and rear suspension of the vehicle, respectively, and x3 is the distance of dead weight from the left end. One may assume that the solution of Eq. (22) is separable in space and time, and the forced solution has the following form,

$$w(x,t) = \sum\limits_{n = 0}^{\infty } {W_{n} } (x)U_{n} (t)$$
(23)

where Wn(x) depends on the spatial position and Un(t) depends on time. Introducing Eq. (23) into Eq. (22), and multiplying each term by Wm(x) and integrating with respect to x = 0, x = l, one may then use the orthogonal property to the modes and obtain

$$\begin{aligned} & \int\limits_{0}^{l} {\rho AW_{n}^{2} } {\text{d}}x\ddot{U}(t) + \omega_{n}^{2} (\rho I\lambda_{n}^{2} - \rho A)\int\limits_{0}^{l} {W_{n}^{2} } {\text{d}}xU(t) \\ & \quad = F_{1} W_{n} (x_{1} ) + F_{2} W_{n} (x_{2} ) + F_{3} W_{n} (x_{3} ) \\ \end{aligned}$$
(24)
$$\begin{aligned} & \rho A\left( {1 - \frac{I}{A}\lambda_{n}^{2} } \right)\int\limits_{0}^{l} {W_{n}^{2} } {\text{d}}x\ddot{U}(t) + \rho A\left( {1 - \frac{I}{A}\lambda_{n}^{2} } \right)\omega_{n}^{2} \int\limits_{0}^{l} {W_{n}^{2} } {\text{d}}xU(t) \\ & \quad = F_{1} W_{n} (x_{1} ) + F_{2} W_{n} (x_{2} ) + F_{3} W_{n} (x_{3} ) \\ \end{aligned}$$
(25)

Equation (25) can be represented in general form as

$$m\ddot{U}(t) + kU(t) = F_{1} W_{n} (x_{1} ) + F_{2} W_{n} (x_{2} ) + F_{3} W_{n} (x_{3} )$$
(26)

where

$$\begin{aligned} m & = \rho A\left( {1 - \frac{I}{A}\lambda_{n}^{2} } \right)\int\limits_{0}^{l} {W_{n}^{2} } {\text{d}}x \\ k & = m\omega_{n}^{2} \int\limits_{0}^{l} {W_{n}^{2} } {\text{d}}x \\ \end{aligned}$$

Now one may write the first frequency modes (Karnopp et al. 2012)

$$\left[ {\rho A\int\limits_{0}^{l} {(1)_{n}^{2} } {\text{d}}x} \right]\ddot{U}_{00} = F_{1} + F_{2} + F_{3} \quad {\text{or}}\quad m\ddot{U}_{00} = F_{1} + F_{2} + F_{3}$$
(27)

which states that the external force accelerates the center of mass of the beam. The other zero frequency mode yields

$$\left[ {\rho A\int\limits_{0}^{l} {\left( {x - \frac{l}{2}} \right)^{2} {\text{d}}x} } \right]\ddot{U}_{00} = F_{1} + F_{2} + F_{3}$$
(28)

or

$$J\ddot{U}_{0} = F_{1} + F_{2} + F_{3}$$

This equation states that the moment of the external forces about the center of mass produces angular acceleration \(\ddot{U}_{00}\), where J is the centroid moment of inertia of beam. The next subsection will present the formulation, which includes structural damping.

2.4 Incorporation of Structural Damping

Now one may consider internal damping (the detailed derivation of internal damping is presented in “Appendix 2” for ready reference) of the beam and Eq. (22) may be rewritten as

$$\begin{aligned} & \rho A\ddot{w}(x,t) - \left( {\rho I\dot{w}^{{\prime \prime }} (x,t)} \right)^{{\prime }} + \left( {EIw^{{\prime \prime }} (x,t)} \right)^{{\prime \prime }} - \mu_{I} I\left( {\dot{w}^{{{\prime \prime \prime }}} } \right)^{{\prime }} \\ & \quad = F_{1} \delta (x - x_{1} ) + F_{2} \delta (x - x_{2} ) + F_{3} \delta (x - x_{3} ) \\ \end{aligned}$$
(29)
$$\begin{aligned} & \rho A\ddot{w}(x,t) - \left( {\rho I\dot{w}^{{\prime \prime }} (x,t)} \right)^{{\prime }} + \omega^{2} (\rho I\lambda^{2} - \rho A) - \mu_{I} I\left( {\dot{w}^{{{\prime \prime \prime }}} } \right)^{{\prime }} \\ & \quad = F_{1} \delta (x - x_{1} ) + F_{2} \delta (x - x_{2} ) + F_{3} \delta (x - x_{3} ) \\ \end{aligned}$$
(30)

where μI is the structural damping coefficient, δ is the Dirac delta function, x1 and x2 are the respective distance from left corner to the front and rear suspension of the vehicle and x3 is the distance of dead weight from the extreme left corner of the vehicle. F1 and F2 are the suspension forces of the front and rear suspension system, respectively, and F3 is the dead weight. The front and rear suspension forces of the vehicle may be expressed as

$$F_{1} = C_{\text{s}}^{\text{f}} (V_{\text{f}} - \dot{w}_{1} ) + K_{\text{s}}^{\text{f}} \left( {\int\limits_{0}^{t} {V_{\text{f}} } {\text{d}}t - w_{1} } \right)$$
(31)
$$F_{2} = C_{\text{s}}^{\text{r}} (V_{\text{r}} - \dot{w}_{2} ) + K_{\text{s}}^{\text{r}} \left( {\int\limits_{0}^{t} {V_{\text{r}} } {\text{d}}t - w_{2} } \right)$$
(32)

where \(C_{\text{s}}^{\text{f}}\) and \(C_{\text{s}}^{\text{r}}\) are, respectively, damping coefficient of the front and rear damper; Vf and Vr are the road velocity input at front and rear wheel, respectively; \(K_{\text{s}}^{\text{f}}\) and \(K_{\text{s}}^{\text{r}}\) are the suspension stiffness of front and rear suspension system, respectively; \(\dot{w}_{1}\) and w1 are the velocity and displacement of front sprung mass; and \(\dot{w}_{2}\) and w2 are the velocity and displacement of rear sprung mass.

One may assume solution similar to Eq. (24); applying Eqs. (23)–(30), one may obtain a solution as

$$\begin{aligned} & \int\limits_{0}^{l} {\rho AW_{n}^{2} } {\text{d}}x\ddot{U}(t) + \omega_{n}^{2} (\rho I\lambda_{n}^{2} - \rho A)\int\limits_{0}^{l} {W_{n}^{2} } {\text{d}}xU(t) - \rho I\lambda_{n}^{2} \int\limits_{0}^{l} {W_{n}^{2} } {\text{d}}x\ddot{U}(t) \\ & \quad - \mu_{I} I\lambda_{n}^{4} \int\limits_{0}^{l} {W_{n}^{2} } {\text{d}}x\dot{U}(t) = F_{1} W_{n} (x_{1} ) + F_{2} W_{n} (x_{2} ) + F_{3} W_{n} (x_{3} ) \\ \end{aligned}$$
(33)

Rearranging Eq. (33), one may obtain

$$\begin{aligned} \rho A\left( {1 - \frac{I}{A}\lambda_{n}^{2} } \right)\int\limits_{0}^{l} {W_{n}^{2} } {\text{d}}x\ddot{U}(t) + \rho A\left( {1 - \frac{I}{A}\lambda_{n}^{2} } \right)\omega_{n}^{2} \int\limits_{0}^{l} {W_{n}^{2} } {\text{d}}xU(t) \hfill \\ - \mu_{I} I\lambda_{n}^{4} \int\limits_{0}^{l} {W_{n}^{2} } {\text{d}}x\dot{U}(t) = F_{1} W_{n} (x_{1} ) + F_{2} W_{n} (x_{2} ) + F_{3} W_{n} (x_{3} ) \hfill \\ \end{aligned}$$
(34)

Finally, Eq. (34) may be represented as

$$m_{n} \ddot{U}(t) + R_{n} \dot{U}(t) + k_{n} U(t) = F_{1} W_{n} (x_{1} ) + F_{2} W_{n} (x_{2} ) + F_{3} W_{n} (x_{3} )$$
(35)

where

$$m_{n} = \rho A\left( {1 - \frac{I}{A}\lambda_{n}^{2} } \right)\int\limits_{0}^{l} {W_{n}^{2} } {\text{d}}x,$$
(36)
$$k_{n} = m\omega_{n}^{2} \int\limits_{0}^{l} {W_{n}^{2} } {\text{d}}x,$$
(37)
$$R_{n} = \mu_{I} I\lambda_{n}^{4} \int\limits_{0}^{l} {W_{n}^{2} } {\text{d}}x$$
(38)

Equations (36–38) show that the rotary inertia affects the dynamics of the beam, which is not being taken in case of Euler beam. However, in this case, the value of \({I \mathord{\left/ {\vphantom {I A}} \right. \kern-0pt} A} \cdot \lambda_{n}^{2}\) is always less than 1. Generally, distributed elements that one may obtain to involve in almost all the system models are lightly damped. The location of structure damping cannot be precisely identified from where it arises. The mechanism of energy dissipation is very typical and complicated; some dissipation takes place because of the plastic deformation and some of the energy radiation from material surfaces. It is not assured in the modeling that individual conventional damper will attach to model for representing the damping of distributed elements. It can happen if the damping includes functionally by incorporating it into individual modes. This is accompanied by merely appending R-elements to the mode oscillators of the modal bond graph. R-element is represented in Eq. (30), where μI is varied from 0.01 to 0.1. The next subsection will present the bond graph modeling to visualize the simulation for different parameters.

3 Bond Graph Modeling of the Truck Chassis Structure

The physical system of a truck chassis (pitch plane model) with front and rear suspension system driven over the random road is presented in Fig. 1. The boundary condition of the structure is taken as free–free. The dynamic disturbance in vertical direction of the vehicle due to random road conditions is the main attraction of this modeling. The longitudinal and lateral dynamics are not considered in the modeling. The bond graph technique is being used as a modeling tool, and the bond graph of the system is created through formulation, which was developed in the previous section. Figure 2 presents the bond graph model of a flexible infinitesimal beam, whereas Fig. 3 shows the bond graph model of a flexible beam with internal damping along with suspension system. The rigid body modes appear simply as inertia elements with associated modal stiffness and damping. The first inertia parameter is the beam mass (M), and second is the beam moment of inertia (Jg). The transformer (-TF-) elements are connected to the rigid body modes, which correctly apply the forces and moments to these elements. The modal masses are taken from Eq. (36), whereas modal stiffness and modal damping can be taken from Eqs. (37) and (38), respectively.

Fig. 2
figure 2

Bond graph model of truck chassis considering internal damping

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Fig. 3
figure 3

Bond graph model of truck chassis considering internal damping

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The system is entirely flexible in transverse and bending, but torsionally rigid. Each 0-junction is attached to each 1-junction with modal components through transformer (TF) elements with moduli equal to appropriate mode function. The transformer function is moduli with shape function (Wn), and the values of this function can be determined from Eq. (16). The attached systems are then appropriately appended to the external 0-junction bonds to form a complete, low-order, very accurate model. Figure 3 shows a general finite mode representation with two input (front and rear) location, where Eq. (26) represents a governing contribution of ith mode.

In the real world, distributed elements have some damping properties, which resist vibration level up to a certain level. No mechanism can quickly identify the value of damping on specific. However, only the quantity of dissipative energy can be determined. It is customary not to attempt to perform detailed modeling of the damping mechanism, but instead to include damping functionally by incorporating its individual modes. This is accomplished by simply appending R-elements to the mode oscillator of the bond graph. The bond graph model of the distributed system including damping modes is presented in Fig. 3. The governing equation of this configuration is mentioned in Eq. (23).

4 Numerical Simulation

Computer simulation can compress the performance of a system over the years into a few minutes of the computer running time. The parameters used for simulation for truck chassis system are presented in Table 1. Simulation models are relatively flexible and can be modified to accommodate the changing environment to the real situation. At present, most of the simulation models are made using differential equations. In this research, a half car flexible model with suspension systems is investigated at different operating speeds on different road profile inputs using bond graph technique and simulators of SYMBOLS Sonata® and MATLAB/Simulink® software are used (Gupta and Rastogi 2016; Hassani 2013; Mukherjee and Samantaray 2006). The bond graph model of the vehicle is simulated for 10 s to obtain different output responses. In total, 1024 records are used in the simulation and error is kept in the order of 5.0 × 10−4. Fifth-order Runge–Kutta Gill method is used in this present work to solve the various differential equations generated through bond graph model.

Table 1 Simulation parameters for flexible chassis system (Rideout and Khan 2010)
Full size table

4.1 Road Inputs

Two types of road profile inputs of the simulation of this system are used, viz. steady-state input and random road input.

4.1.1 Sine Wave Input

The steady-state input has been taken in a sinusoidal form, and the following expression is used for simulation (Hagedorn and DasGupta 2007):

$$V_{\text{input}} = A_{\text{jump}} *\omega *\sin (\omega *t),\quad {\text{for}}\,{\text{front}}\,{\text{wheel}}$$
(39)
$$V_{\text{input}} = A_{\text{jump}} *\omega *\sin \left( {\omega *(t + d/v)} \right),\quad {\text{for}}\,{\text{rear}}\,{\text{wheel}}$$
(40)

where Ajump is the amplitude of vibration and ω is the excitation frequency.

4.1.2 Random Road Condition

A random road profile is generated according to the International Organization for Standardization (ISO 8608). It gives a depiction of the road profile through estimation of the PSD of the vertical displacements Gd, as a function of spatial frequency \(n(n = \varOmega /2\pi \left( {\text{cycles}} \right)/{\text{m}})\) and also of angular spatial frequency Ω. ISO 8608 introduces a classification, which is evaluated in accordance with conventional values of spatial frequency n0 = 0.1 cycles/m and angular spatial frequency Ω0 = 1 cycles/m. There are eight classes of roads mainly identified from classes H1 to H8 according to the values of Gd(n) and Gd(Ω) established in ISO 8608, which are shown in Table 2 (Meirovitch 1980).

Table 2 ISO 8608 values of Gd(n0) and Gd(Ω0)
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In the simulation, ISO 8608 provides the roughness of the road surface profile, which may be stated using the following equations:

$$G_{\text{d}} (n) = G_{\text{d}} (n_{0} )\left( {\frac{n}{{n_{0} }}} \right)^{ - w}$$
(41)
$$G_{\text{d}} (\varOmega ) = G_{\text{d}} (\varOmega_{0} )\left( {\frac{n}{{n_{0} }}} \right)^{ - w}$$
(42)

where w denotes the waviness and its value is taken to be 2 in this case. In this research, random road inputs have been developed by taking into consideration the PSD of vertical displacement Gd as a function of spatial frequency n.

Beginning with a distributed road profile to specify the value of spatial frequency n, looped within a frequency band Δn, the PSD function value is represented through the following expression as

$$G_{\text{d}} (n) = \mathop {\lim }\limits_{\Delta n \to 0} \left( {\frac{{\psi_{x}^{2} }}{\Delta n}} \right)$$
(43)

where the mean square value is denoted by \(\psi_{x}^{2}\) which signifies the component of the signal for spatial frequency n contained by the frequency band Δn. Accordingly, the signal of road profile is discretized, and thus, it is characterized by a series of elevation points along with evenly spaced ones. Here, road profile length is denoted by L, whereas the sampling interval is denoted by B. Further, the maximum value of sampling spatial frequency c is 1/B, the maximum effective sampling spatial frequency (neff) is nmax/2 and the discretized spatial frequency values ni are equally spaced under the frequency domain, with an interval of (Δn) as 1/L. So herein, generic spatial frequency value ni can be regarded as i·Δn and Eq. (43) may be written in the discrete form as

$$G_{\text{d}} (n) = \frac{{\psi_{x}^{2} (n_{i} ,\Delta n)}}{\Delta n} = \frac{{\psi_{x}^{2} (i \cdot \Delta n,\Delta n)}}{\Delta n}$$
(44)

where i varying from 0 to N as \(\frac{{n_{\hbox{max} } }}{\Delta n}\), one may obtain the road profile by a simple harmonic function

$$h(x) = A_{i} \cos (2\pi \cdot n_{i} \cdot x + \phi ) = A_{i} \cos (2\pi \cdot i \cdot x \cdot \Delta n + \phi )$$
(45)

where the amplitude is denoted by Ai, spatial frequency is denoted by ni and phase angle is φ. One may generate a harmonic signal through mean square value

$$\psi_{x}^{2} = \frac{{A_{i}^{2} }}{2}$$
(46)

Therefore,

$$G_{\text{d}} (n_{i} ) = \frac{{\psi_{x}^{2} (n_{i} )}}{\Delta n} = \frac{{A_{i}^{2} }}{2 \cdot \Delta n}$$
(47)

It has been shown in several works that the development of an artificial road profile using Eq. (45) is only possible if the PSD function of vertical displacements is well known, presuming a random phase angle φi following a uniform probabilistic distribution within range 0 to 2π. The artificial profile can be given as

$$ \begin{aligned}h(x) &= \sum\limits_{i = 0}^{N} {A_{i} \cos (2\pi \cdot n_{i} \cdot x + \phi_{i} )} \\ &= \sum\limits_{i = 0}^{N} {\sqrt {2 \cdot \Delta n \cdot G_{\text{d}} (i \cdot \Delta n) \cdot } A_{i} \cos (2\pi \cdot i \cdot x \cdot \Delta n + \phi )}\end{aligned} $$
(48)

Using Eq. (47) in Eq. (45), a random road profile can be generated according to ISO classification by the following equation,

$$h(x) = \sum\limits_{i = 0}^{N} {\sqrt {\Delta n.} \cdot 2^{k} \cdot 10^{ - 3} \cdot \left( {\frac{{n_{0} }}{i \cdot \Delta n}} \right)\cos (2\pi \cdot i \cdot x \cdot \Delta n + \phi )}$$
(49)

where x is the abscissa variable from 0 to L; Δn is 1/L; nmax is taken as 1/B; N is \(\frac{{n_{\hbox{max} } }}{\Delta n}\) or L/B; where L = 250, N = 100; and constant is denoted as k, depending on ISO road profile classification. It has also assumed integers augmenting from 3 to 9, which corresponds to the profiles from classes H1 to H8 (as shown in Table 3); here, k is assumed to take the value 3 corresponding to the class H1 road profile and Gd(n0) is taken to be 32, where n0 is 0.1 cycles/m and φi is the random phase angle following an uniform probabilistic distribution within range 0 to 2π. In this work, four road conditions (H1-H4) are considered, which is shown in Figs. 4, 5, 6 and 7.

Table 3 k values for ISO road roughness classification
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Fig. 4
figure 4

Random road profile of H1 road category at 40 kmph, 60 kmph, 80 kmph, 90 kmph and 100 kmph

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Fig. 5
figure 5

Random road profile of H2 road category at 40 kmph, 60 kmph, 80 kmph, 90 kmph and 100 kmph

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Fig. 6
figure 6

Random road profile of H3 road category at 40 kmph, 60 kmph, 80 kmph, 90 kmph and 100 kmph

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Fig. 7
figure 7

Random road profile of H4 road category at 40 kmph, 60 kmph, 80 kmph, 90 kmph and 100 kmph

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The simulation of different road categories is obtained by MATLAB/Simulink of the previously discussed ISO 8608 random input. This road profiles are simulated for five different speeds, viz. 40 kmph, 60 kmph, 80 kmph, 90 kmph and 100 kmph. Figure 4 shows H1 road condition, where unevenness of the random-type road with maximum magnitude is 0.03 m in an upward direction at 40 km/h and − .035 m in a downward direction at 100 km/h. Similarly, H2, H3 and H4 road categories have a maximum magnitude 0.06 m, 0.12 m and 0.24 m, respectively, in an upward direction at 40 km/h and − 0.65 m, − 0.12 m and − 0.24 m, in a downward direction at 100 km/h.

5 Results and Discussion

The model is exhibited first on sine wave input from start to end and runs for 20 s with zero initial conditions. The amplitude of sine wave is 0.01 m, and the input frequency is taken 50 Hz. In Fig. 8, the magnitude of the vertical acceleration is 0.8 m/s2 in un-damped condition, whereas acceleration amplitude has been reduced to 0.7 m/s2 in damped condition. Thus, the magnitude of vibration has been reduced from 12 to 15% due to the impact of structural damping of the vehicle chassis. Similarly, in Fig. 9, the amplitude of pitching acceleration has been reduced up to 20% due to the influence of structural damping. However, frequencies in both conditions have almost similar values. The dead load on the center of chassis has been taken around 500 N, which lies in the downward direction.

Fig. 8
figure 8

Vertical response of vertical acceleration of chassis at steady-state input

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Fig. 9
figure 9

Response of pitch acceleration of chassis at steady-state input

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The RMS acceleration responses of truck chassis at the first three modes at sine wave condition are shown in Fig. 10a–c. In these figures, the un-damped condition of the beam is superimposed over damped condition. The frequencies of the first three modes are 34.4 Hz, 86.89 Hz and 174.8 Hz, respectively. Figure 10a–c shows that the acceleration magnitude is continuously reduced, whereas maximum acceleration is found at the first mode, and minimum acceleration is in the third mode. The influence of internal damping is presented in terms of RMS acceleration response, whereas the amplitude has reduced due to the effect of internal damping at the same frequency.

Fig. 10
figure 10

RMS acceleration response of different modes of chassis under sine wave condition: a first mode = 34.4 Hz; b second mode = 86.89 Hz; c third mode frequency = 174.8 Hz

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The ride comfort analysis may be evaluated through RMS acceleration of the vehicle chassis. Figures 11, 12, 13 and 14 present the RMS acceleration response of the truck CG for four different classes of roads (H1–H4) as prescribed by ISO 8608. The H1 is the best class, and H4 represents the worst road class in these four classes. The model is simulated for each road class at different speeds ranging from 40 to 100 kmph with and without consideration of the internal damping of the chassis. Figures 11, 12, 13 and 14 show that the magnitude of vertical acceleration increases with vehicle speed and with road class. Figure 11a demonstrates the truck response neglecting the chassis damping, where maximum RMS acceleration magnitude is observed as 0.65 m/s2 at 100 km/h and 0.25 m/s2 at 40 km/h on H1 class road. This peak acceleration magnitude gets reduced up to 0.58 m/s2 at 100 km/h and 0.22 m/s2 at 40 km/h with due consideration to chassis structural damping, as shown in Fig. 11b. A similar kind of result has been noticed in other plots shown in Figs. 12, 13 and 14. Thus, it is evident from the above results that internal structural damping of the base structure of heavy vehicle plays an important role and it needs to be included in the dynamical model of the vehicle to get the true dynamic behavior of the vehicle.

Fig. 11
figure 11

a Response of chassis acceleration at various random road inputs in H1 road as per ISO 8608. b Response of chassis acceleration consideration of internal damping at different random inputs in H1 road condition as per ISO 8608

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Fig. 12
figure 12

a Response of chassis acceleration at various random road inputs in H2 road as per ISO 8608. b Response of chassis acceleration consideration of internal damping at different random inputs in H2 road condition as per ISO 8608

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Fig. 13
figure 13

a Response of chassis acceleration at various random road inputs in H3 road as per ISO 8608. b Response of chassis acceleration consideration of internal damping at different random inputs in H3 road condition as per ISO 8608

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Fig. 14
figure 14

a Response of chassis acceleration at various random road inputs in H4 road as per ISO 8608. b Response of chassis acceleration consideration of internal damping at different random inputs in H4 road condition as per ISO 8608

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It has been observed that from the RMS curves that the H1 road condition is perfect for driving the vehicle for long duration, whereas it is not advisable to drive vehicles for long duration on H4 road condition. Indian roads lie between H2 and H3 conditions. So, a driver can comfortably drive the given vehicle at 60–80 km/h in single stretch without a break.

6 Conclusions

Numerous models have been constructed for the heavy vehicle system, but most of the models are developed with the lumping process. In this work, an analytical framework for truck chassis has been developed through classical mechanics and through extended Lagrangian approach. The following conclusion has been drawn from present research work:

  • In the classical mechanics approach, the idea of lumped–distributed modeling has been explored within the framework of bond graphs.

  • The Lagrange variation of truck chassis, including flexural behavior has been constructed, where it was integrated with lumped suspension system. However, the structural damping effects were also involved in terms of fifth-order partial differential term in generalized equation of motion. The detailed derivation for fifth-order differential term is presented in “Appendix 1.”

  • The integrated bond graph model of flexural beam was the extended form of the analytical model. Results have been presented under the two types of road conditions, i.e., sine wave input and random road input. In sine wave road conditions, the vertical response of flexural beam is reduced up to 12–15% and pitch response of beam is reduced up to 20% due to the structural damping effect. In random road conditions, four types of road category (H1, H2, H3, H4) have been used at four different speeds (40 kmph, 60 kmph, 80 kmph and 100 kmph).

  • It is depicted from the result that maximum acceleration at maximum speed (100 kmph) of chassis has been increased up to 0.5 m/s2 in H1–H2 conditions, 1 m/s2 in H2–H3 conditions and 2.25 m/s2 in H3–H4 conditions. Similarly, the solution has been extended for structural or internal damping, whereas maximum acceleration at maximum speed has been reduced up to 10%, 5%, 7% and 5% (approx.) in H1, H2, H3 and H4 road conditions, respectively.