In automobile wishbone suspension using Simmechanic and Simulink

In a mass spring damper systems accurate control of motion (such as position, velocity) is a fundamental concern to control engineers. Therefore, the position control attempts to shape the dynamic of the mass while achieving the constraints imposed by the force positioning the mass. In general, the control problems involve finding suitable mathematical models that describe the dynamic behaviour of the physical mass spring dampers (MSD) model to permit suitable controller design and allow corresponding control strategies to realize the expected system response and performance. Mass spring damper systems (MSD) have  wide range of applications  like robot manipulator control1,2,vehicle suspension systems for shock absorption in automobiles 3,4,5,6,7. Recently, MSD systems are on increasing demand in hybrid vehicle suspension to increase passenger ride comfort and vehicle stability over cracks and uneven pavement.Tarik et al  2 developed a mass spring damper model with MATLAB graphical user interface that permit different choice of control strategies on the model by changing the model parameters. Sivák and Hroncová 8 presented equations of motion (EOM) of a mechanical system with two degree of freedom in MATLAB/Simulink using state space and transfer function. The conclusion of their work is that Newton’s law and Lagrange’s equation resulted in the same solution. In 9, the author presented a model for quarter car automobile wishbone suspension using Simmechanic and Simulink and implemented proportional-integral-derivative (PID) control strategy to minimize the vertical body acceleration. In 10,11, the authors developed mass spring damper models based on Newton’s law of motion to derive state-space model for numerical computation with the aids of MATLAB and Simulink.                Author in 12 examined the effect of position of the damper in systems with multi-degree of freedom. Their research work concluded that displacement of oscillator that experience the effect of force show more energy absorption for low displacement. In 3, a Simscape and Simulink based model are  developed to investigate the effect of seat suspension  on vehicle performance. In 13,14, the authors presented mathematical modeling of a mass spring damper system in MATLAB and Simulink. The author in  15, presented control of coupled mass spring damper system using polynomial structures approach. Lee et al 1 proposed novel backstepping and model reference adaptive control for position control of chained multiple mass spring damper systems.The objectives of this paper is to establish a mathematical model that represent the dynamic behaviour of a coupled mass spring damper systems and effectively control the mass position using both Simulink and Simelectronics as simulation tools. 1. Methodology We derived the equations of motion (EOM) of  a coupled mass spring damper systems using second-order, ordinary differential equations and to stimulate dynamic accurately16  the Lagrange’s equation was adopted. The motivation for chosen Lagrange’s equation over Newton’s law or D’Alembert principle is that is quite unwieldy for large complex systems and time consuming to work out all forces acting on the body. The mathematical model is formulated based on energy property of Lagrange approach and the control strategy, Simulink and Simelectronics simulations are expanded on the derived mathematical model. However, the langrage’s equation does not improvise for dissipative (damping) force in the mechanical system, hence, Rayleigh’s dissipation function is introduced into Lagrange’s equation to account for dissipative force in the model and we refer to this as augmented Lagrange’s equation. In order to describe the physical motion of a coupled mass spring damper systems, we need to choose a set of variables or coordinates which are often referred to as generalized coordinates. Thus, the displacement of the masses are chosen as the generalized coordinates. 2.1. Mathematical modeling The dynamic of a mass spring damper systems with two degree of freedom (DOF) movement is explicitly derived based on Lagrange’s equation to expound the problems involved in dynamic modeling. Figure 1 depict  a coupled mass spring damper systems, where two masses m1 and m2 are linked to a parallel spring-damper configuration with spring stiffness coefficients k1 and k2 and viscous damping coefficients b1 and b2  for mass m1 and m2 respectively. The force f acts on mass 2 and the energy is transmitted to mass 1 via the springs and some part of energy is absorbed by the dampers