The dynamic accuracy and stability of machine-tool hydraulic copying systems are investigated by formulating a mathematical model of a copying unit, which takes into account the following features:​ a) dynamic cutting force and its interaction with the velocity response of the system, b) dynamic behavior of the copying slide with the dry friction in its mountings, c) dynamic behavior of the control valve and the action of the flow forces, and d) dynamic behavior of the stylus.

The time domain model of the kinematic input due to the template configuration which gives the actual motion of the stylus during copying is used as the input function in the analysis. This procedure makes both analysis and results applicable to all types of machine-tool hydraulic copying systems, including electro-hydraulic and numerically-controlled devices.

The governing equations of the system are non-dimensionalized and simulated on an analog computer to study the effects of various influencing parameters on the dynamic accuracy and stability. For this purpose, a dynamic accuracy criterion which is a function of the manufacturing tolerance of the produced parts is developed. The comparison of theoretical results with experimental data have been achieved by comparing the theoretical results with actual test data supplied by the manufacturers. Close agreement is observed.

The results indicate that the stability of copying depends to a great extent on the amount of dry friction, inertia, size, and stiffness of the stylus, the supply pressure and the cross-sectional area of the piston. The results also show that the system accuracy is increased by decreasing the inertia and increasing the stiffness of the stylus. Increasing, the supply pressure and the cross-sectional area of the piston also improves the dynamic accuracy of the system.

Further results in this investigation are outlined as:​

• dry friction in the system increases the stability but decreases the dynamic accuracy.
• dynamic accuracy increases by increasing the kinematic gain K_G but decreases the stability.
• leakage flow increases the stability at the cost of a decrease in the dynamic accuracy.
• increasing the angle y and decreasing the velocity v of the machine-tool slide increases the dynamic accuracy.
• decrease of the exhaust pressure increases the dynamic accuracy and decreases the stability.
• from the stability point of view, the dry friction in the system should be higher for finishing operation than for roughing operation.

The results also provide some useful informations on:​

• control of stability of existing devices.
• optimization of existing copying devices for maximum accuracy under stable operation.

Plots are developed to be used as a guide for the design and operation of hydraulic copying systems.