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INTRODUCTION

This Data Item presents the analytical methods used to estimate numerical values of reaction forces at joints that connect links of single degree-of-freedom, planar, multibar linkage mechanisms. Force analysis is an essential feature of the design of linkages used in industrial machinery because it can lead to improved reliability combined with higher operating speeds and loads.

It is advisable, during the design of a new mechanism, to obtain estimates of the joint forces^† once the link lengths have been determined. Provisional values of cross-sectional geometry and a suitable material should be chosen for each moving link. Force analysis gives provisional values of joint forces which are then used to estimate the stresses in individual links and the deflections of important points on those links. The proposed cross-sectional geometry and material may then be altered and the joint forces re-determined in an iterative process until an acceptable design is obtained. Also estimates of joint forces are used to determine appropriate bearing types and sizes. Values of frame shaking force and frame shaking moment are essential when designing the fixed machine frame and when deciding whether it is necessary to balance a linkage. Force values are used to compare the performance of alternative linkage configurations and to evaluate the effect of introducing different balancing techniques (see References 9, 10 and 11).

Traditionally, values of the necessary kinematic and dynamic quantities are obtained by a laborious graphical approach. Modern methods are based on the derivation of numerous equations describing the kinematic relationships and the equilibrium conditions followed by numerical solution of these equations using a computer. Three general strategies for estimating joint forces are described in this Item.

(a) Static force analysis

Inertia is ignored and the reactions at the joints due only to the external force are determined. Each joint force is expressed as a function of the position of the input link and the linkage is assumed to be stationary at each position. Static force analysis is suitable only for the analysis of very slow moving linkages.

(b) Kinetostatic force analysis

The reactions at the joints due to inertia and external forces are determined. Each joint force is expressed as a function of the position of the input link and the motion of the input link is assumed to be known. A commonly-used assumption is that the mechanism is driven via an input crank rotating at constant angular velocity.

(c) Dynamic force analysis

The reactions at the joints due to inertia and external forces are determined. Each joint force is expressed as a function of time and the driving force or torque is assumed to be known as a function of time, input link position or input link velocity.

The development of rigid-body mathematical models of linkages using these strategies is demonstrated in this Item. The range of application and the advantages and limitations of each method are stated. The wide variety of linkage configurations used in industrial machinery and the different loading conditions of each machine make the presentation of generic design data impractical. Consequently the aim of this Data Item is to provide a simple framework for the development of computer-based methods for analysis of linkages in particular applications. Section 11 contains the equations required to determine the joint forces by kinetostatic force analysis for commonly-used four-bar linkages. A computer program implementing kinetostatic force analysis of four-bar planar linkages is included in Section 12.

^† See Section 2.2 for definitions