In this obituary, we present the life and achievements of an outstanding Polish scientist, Professor Jan Kruszewski-Majewski, who passed away in 2012. Professor Kruszewski-Majewski was one of the pioneers in introducing modern numerical methods in mechanics. He created and developed an original, Polish method of modelling called the rigid finite element method. He was not only an outstanding scientist, but also a person of great charisma and moral rectitude.

In the paper, the authors describe and solve the problem of optimum control of selected vibration forms in mechanical systems. Two illustrative examples have been used to present the procedure for determination of the optimum controller coefficients.

In the first example, a simplified mechanical system is considered, while in the second one – a rotor with magnetic bearing. In both cases, the integral performance indices have been defined in order to minimize the vibration level at selected points of the structures.

The system with the magnetic bearing is structurally unstable. For this reason, the authors present the way of finding the weight coefficients of integral performance index for unstable, multi-degrees-of-freedom system. In that way, the selected modal forms attain the previously assumed dynamic properties and the performance index takes the minimum value. The results of numerical analysis show that the proposed way is efficient and makes it possible to control selected forms of vibration in the system.

An original method of skeletal system modelling is presented in detail. Using DICOM images obtained from CT and PET tests, shell models of nine bones were created (humerus, radius, ulna, scapula, clavicle, femur, tibia, fibula, pelvis). Two methods of bone behaviour are also proposed, the first method treating the bone as a solid structure and the second method treating the bone as a complex porous structure. The behaviour of model parts is numerically examined by using the finite element method.

The paper presents a spatial model of the satellite antenna with an arbitrary number of flexible arms. Such a system is an example of an open kinematic chain with a tree-like structure. The modification of the rigid finite element method is used to discretise flexible links. The equations of motion are derived from the Lagrange equations and the motion of the system is described using joint coordinates and homogenous transformations. Numerical simulations have been carried out to analyse how the method of extending the arms influences the dynamics of the system.