TUTorial

PrinciplesIn the engineering world, many complicated mechanisms are represented and studied by using simple models. One such example is the wheel and suspension arm of a vehicle.

In this particular case, the modal behaviour of this simple suspension system can be modelled as a 2 DOF spring-mass system.

Once the parameters representing the model (masses, springs and displacements) have been defined, applying Newton’s second law to the FBD in Fig. 2, the equations of motion are:

The equations above describe a second order ordinary differential equation system, which can be expressed in matrix form as:

where the mass and stiffness matrices are:

The damping matrix C is not required since this tutorial is considering an undamped system.

The natural frequencies of vibration will be obtained by resolving the classical eigenvalue problem defined as:

followed by a simple square root calculation:

Practical CaseIn order to understand the modal behaviour of an automotive suspension system, similar to the one displayed in Fig. 1, it has been modelled as a 2 DOF spring-mass system, where K1 represent the stiffness of the tyre, M1 the wheel mass, K2 represent the stiffness of the suspension arm and M2 the car mass. The values for each parameter are 20 N/m, 5 Kg, 8 N/m and 150 Kg respectively. The natural frequencies of vibration

are required to later improve handling characteristics of the suspension system.

SolutionThe mass and stiffness matrices are

The eigenvalue problem is defined as

hence the eigenvalues are

and the natural frequencies are

HyperMath implementationUsing HyperMath, a function will be created to perform all the calculations required to obtain the natural frequencies of a model as described in the Practical Case. Additionally, an MBD model built in MotionView will be provided and the files corresponding to the modal analysis performed by MotionSolve.

Phase 1 – Create the mathematical model in HyperMathWithin the HyperMath environment, a function will be created in order to perform all the calculations required to obtain the natural frequencies of the system. The aim of this phase is to get familiar with the easy scripting language of HyperMath.

Phase 2 – Compare results with MotionView/MotionSolveA MotionView model and the files from MotionSolve’s frequency analysis are also included in the download in order to compare the pure mathematical approach and the more visual implementation.

HyperMath – MotionSolve ResultsThe results provided by both approaches are the same but it would be interesting to further implement the model with damping or even better, generate a full model of four suspension system as if it was a full vehicle analysis.

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Fig. 1 Vehicle Suspension System and 2 DOF Spring-Mass model

Fig. 2 Vehicle Suspension System FBD