Mechanical Linkages – Four-Bar Mechanisms and Kinematic Analysis

Mechanical linkages are assemblies of rigid links and joints used to transmit or transform motion and force. They are fundamental to mechanisms such as wiper systems, clamps, folding mechanisms, machinery, actuators and motion conversion systems.

This page covers planar linkages, degrees of freedom and mobility, Grashof's criterion, four-bar mechanical advantage, Freudenstein's equation, velocity analysis, acceleration analysis and Coriolis acceleration in sliding members.

For guidance on choosing between linkages, cams, ratchets, indexing mechanisms and other motion systems, see the Mechanism Help and Selection Guide.

Contents

• Introduction to Linkages
• Planar, Spatial and Spherical Mechanisms
• Degrees of Freedom and Mobility
• Grashof's Criterion
• Mechanical Advantage of a Four-Bar Linkage
• Freudenstein's Equation
• Velocity Analysis of Linkages
• Acceleration Analysis of Linkages
• Sliding Block on a Rotating Link

---

Introduction to Linkages

A link is a nominally rigid body connected to other bodies through joints for the purpose of transmitting motion or force. One link is normally fixed relative to the reference frame and forms the frame or ground link of the mechanism.

Examples of linkage applications include garage-door mechanisms, vehicle windscreen-wiper mechanisms, gear-selection mechanisms, clamps, suspension systems, folding mechanisms and machinery.

Links are connected by kinematic pairs. Lower pairs have surface contact between the members. Common examples include:

• revolute or pin joints;
• prismatic or sliding joints;
• screw pairs;
• cylindrical pairs;
• spherical pairs.

Higher pairs involve contact nominally along a line or at a point. Cam-follower and gear-tooth contacts are familiar examples.

The following diagram illustrates examples of kinematic pairs and their associated degrees of freedom.

Kinematic Analysis and Synthesis

Kinematic analysis determines the motion of an existing mechanism from its geometry and specified inputs such as position, angular velocity and angular acceleration.

Kinematic synthesis is the process of selecting the mechanism type and its dimensions so that a required motion or set of positions is obtained.

Analysis therefore asks how a defined mechanism moves, while synthesis asks what mechanism geometry is required to produce a specified motion.

---

Planar, Spatial and Spherical Mechanisms

Planar Mechanisms

A planar mechanism is one in which the trajectories of all points lie in planes parallel to a fixed plane. The axes of revolute joints are therefore parallel to one another and perpendicular to the plane of motion, while prismatic joints translate in directions lying in that plane.

Many practical mechanisms are designed as planar systems because their geometry, manufacture and analysis are comparatively straightforward. A planar mechanism containing only revolute and prismatic lower pairs is commonly described as a planar linkage.

Spatial Mechanisms

A spatial mechanism has genuinely three-dimensional relative motion and is not restricted to parallel planes. Spatial mechanisms may contain revolute, prismatic, cylindrical, spherical, screw or other joint types.

Their analysis and synthesis are generally more involved than for planar mechanisms. Detailed spatial mechanism analysis is outside the scope of this page.

Spherical Mechanisms

In a spherical mechanism, points on the links move on concentric spherical surfaces and the axes of the revolute joints intersect at a common stationary point. The mechanism can therefore be analysed in terms of motion on a sphere.

Detailed spherical linkage analysis is not considered further on this page.

---

Degrees of Freedom and Mobility

The mobility of a mechanism is the number of independent parameters required to define its configuration. It is also commonly called the number of degrees of freedom.

A free rigid link moving in a plane has three degrees of freedom: translation in the x direction, translation in the y direction and rotation about an axis perpendicular to the plane.

If a planar assembly contains n links including the fixed link, the unconstrained moving links initially provide:

$$M=3\left(n-1\right)$$

A one-degree-of-freedom lower pair such as a revolute or prismatic joint imposes two independent constraints. A two-degree-of-freedom planar pair imposes one constraint.

For a planar mechanism, the Kutzbach mobility criterion may therefore be written:

$$M=3\left(n-1\right)-2j_1-j_2$$

where:

• M = mobility or degrees of freedom;
• n = total number of links including the fixed link;
• j₁ = number of joints allowing one relative degree of freedom;
• j₂ = number of joints allowing two relative degrees of freedom.

A correctly constrained arrangement with zero mobility behaves as a structure. A mechanism with mobility one normally requires one independent input to define its configuration, while a mechanism with mobility two requires two independent inputs.

---

Grashof's Criterion for Four-Bar Linkages

When a four-bar linkage is driven continuously by a motor or other rotary source, it is often necessary to determine whether a link can rotate through a complete revolution.

For a planar four-bar linkage, let:

• s = length of the shortest link;
• l = length of the longest link;
• p and q = lengths of the remaining two links.

The Grashof condition is:

$$s+l\le p+q$$

When:

$$s+l<p+q$$

the linkage is a Grashof linkage and at least one link can rotate fully relative to the other links. The actual motion obtained after one link is fixed depends upon which link is selected as the frame.

Four-Bar Inversions

For a Grashof four-bar linkage:

• if the shortest link is fixed, the two adjacent links can rotate completely, giving a double-crank or drag-link mechanism;
• if a link adjacent to the shortest link is fixed, the shortest link can rotate completely while the opposite link oscillates, giving a crank-rocker mechanism;
• if the link opposite the shortest link is fixed, the two links adjacent to the frame rock, giving a double-rocker inversion.

If:

$$s+l>p+q$$

the chain is non-Grashof and no link can make a complete revolution relative to the others; with any link fixed the mechanism behaves as a double rocker.

The equality condition:

$$s+l=p+q$$

is the limiting or change-point case. At certain configurations the links can become collinear, producing a kinematic singularity. Practical mechanisms operating close to this condition require careful consideration of clearances, inertia, friction and the intended branch of motion.

---

Mechanical Advantage of a Four-Bar Linkage

For an ideal four-bar linkage, neglecting friction and inertia, instantaneous mechanical advantage can be expressed as the ratio of output torque to input torque.

Using the notation in the diagram, let:

• b = length of the input or driver link;
• c = length of the coupler;
• d = length of the output or driven link;
• α = angle between the input link and coupler;
• β = angle between the coupler and output link.

The instantaneous angular-velocity relationship is:

$$\frac{{\omega }_{\text{out}}}{{\omega }_{\text{in}}}=\frac{b\sin \alpha }{d\sin \beta }$$

For an ideal loss-free linkage, instantaneous input and output powers are equal in magnitude. Therefore:

$$\frac{T_{\text{out}}}{T_{\text{in}}}=\frac{d\sin \beta }{b\sin \alpha }$$

Thus the mechanical advantage varies continuously with linkage position. It depends upon both the link lengths and the instantaneous angular geometry.

Toggle Positions

When the input link and coupler approach a collinear position, α approaches 0° or 180° and sin α approaches zero. In the ideal kinematic model the torque mechanical advantage therefore tends towards infinity.

Toggle geometry is used deliberately in mechanisms such as clamps because a large output force can be obtained for a relatively small input torque near the toggle position.

Transmission Angle

The angle β between the coupler and output link is commonly called the transmission angle. As sin β approaches zero, the ability of the coupler force to produce output torque decreases and the mechanism approaches a poor force-transmission configuration.

For mechanisms intended to transmit substantial force or torque, it is generally desirable to keep the transmission angle reasonably close to 90° throughout the working range. The acceptable limits depend on the application, loading, friction, bearing design, speed and required force transmission rather than on a universal single limiting angle.

---

Freudenstein's Equation

Freudenstein's equation provides an algebraic relationship between the input and output angles of a four-bar linkage and its link lengths. It is useful both for position analysis and for dimensional synthesis.

Using the link and angle convention shown in the diagram, the vector loop is:

$${\mathbf{l}}_1+{\mathbf{l}}_2+{\mathbf{l}}_3+{\mathbf{l}}_4=0$$

Resolving horizontally:

$$l_1\cos {\theta }_1+l_2\cos {\theta }_2+l_3\cos {\theta }_3+l_4\cos {\theta }_4=0$$

Resolving vertically:

$$l_1\sin {\theta }_1+l_2\sin {\theta }_2+l_3\sin {\theta }_3+l_4\sin {\theta }_4=0$$

With the fixed link oriented so that θ₁ = 180°:

$$-l_1+l_2\cos {\theta }_2+l_3\cos {\theta }_3+l_4\cos {\theta }_4=0$$ $$l_2\sin {\theta }_2+l_3\sin {\theta }_3+l_4\sin {\theta }_4=0$$

Eliminating θ₃ gives:

$$K_1\cos {\theta }_2+K_2\cos {\theta }_4+K_3=\cos \left({\theta }_2-{\theta }_4\right)$$

where:

This is Freudenstein's equation for the angle and link convention used above. Different texts may use different link numbering or angle directions, which changes the apparent signs and definitions of the constants.

For synthesis, specified pairs of input and output angles can be substituted into the equation. Three precision-position pairs provide three simultaneous equations for the three constants, from which suitable link-length ratios can be obtained.

---

Velocity Analysis of Linkages

For a rigid link AB rotating in a plane, the relative velocity of B with respect to A is perpendicular to the link:

$${\mathbf{v}}_{B/A}=\mathbf{\omega }\times {\mathbf{r}}_{B/A}$$

For plane motion its magnitude is:

$$v_{B/A}=\omega \mathrm{AB}$$

Four-Bar Velocity Diagram

For the four-bar linkage shown below, A and D are fixed points. The velocity of B is perpendicular to AB and the velocity of C relative to D is perpendicular to CD.

The vector relationship for point C may be written:

$${\mathbf{v}}_C={\mathbf{v}}_B+{\mathbf{v}}_{C/B}$$

and, because D is fixed:

$${\mathbf{v}}_C={\mathbf{v}}_{C/D}$$

A graphical velocity construction can therefore be made as follows:

• A and D are fixed, so their velocity points coincide at the origin of the velocity diagram.
• The velocity of B is drawn perpendicular to AB with magnitude ω_ABAB.
• From the velocity point for B, draw a line perpendicular to BC representing the possible direction of v_C/B.
• From the fixed velocity origin, draw a line perpendicular to CD representing the possible direction of v_C/D.
• The intersection determines the velocity point for C.
• The velocity of another point P fixed on the coupler can be obtained from the geometry of the velocity diagram.

Sliding Block on a Rotating Link

Consider a block B sliding along a link rotating about A. Let B' denote the point on the rotating link instantaneously coincident with the block.

The absolute velocity of the block is the vector sum of the velocity of the coincident point on the rotating link and the block's sliding velocity relative to the link:

$${\mathbf{v}}_B={\mathbf{v}}_{B\prime }+{\mathbf{v}}_{\text{rel}}$$

where:

$${\mathbf{v}}_{B\prime }=\mathbf{\omega }\times \mathbf{r}$$

---

Acceleration Analysis of Linkages

The acceleration of one point on a rigid link relative to another contains tangential and normal components.

For point B on a rigid link rotating relative to point A:

$${\mathbf{a}}_{B/A}=\mathbf{\alpha }\times {\mathbf{r}}_{B/A}+\mathbf{\omega }\times \left(\mathbf{\omega }\times {\mathbf{r}}_{B/A}\right)$$

For plane motion the magnitudes are:

Four-Bar Acceleration Diagram

For a four-bar linkage with fixed pivots A and D, the acceleration of B is first obtained from the known angular velocity and angular acceleration of link AB.

For the coupler BC:

$${\mathbf{a}}_C={\mathbf{a}}_B+{\mathbf{a}}_{C/B}$$

The normal component of the acceleration of C relative to B has magnitude:

$$a_{n,C/B}={\omega }_{BC}^2\mathrm{BC}$$

and is directed from C towards B. Its tangential component is perpendicular to BC and has magnitude:

$$a_{t,C/B}={\alpha }_{BC}\mathrm{BC}$$

Similarly, because D is fixed, the normal component of C relative to D is:

$$a_{n,C/D}={\omega }_{CD}^2\mathrm{CD}$$

directed from C towards D. The corresponding tangential component is perpendicular to CD.

The intersection of the appropriate tangential construction lines in the acceleration polygon determines the acceleration of C.

Once the acceleration diagram for the coupler is known, the acceleration of another point P fixed on the same coupler can be determined from the rigid-body geometry.

---

Acceleration of a Sliding Block on a Rotating Link

When a block slides along a link that is itself rotating, its acceleration contains an additional Coriolis term.

For a moving reference frame attached to the rotating link, the absolute acceleration of the block can be written:

$$\mathbf{a}={\mathbf{a}}_O+\mathbf{\alpha }\times \mathbf{r}+\mathbf{\omega }\times \left(\mathbf{\omega }\times \mathbf{r}\right)+2\mathbf{\omega }\times {\mathbf{v}}_{\mathrm{rel}}+{\mathbf{a}}_{\mathrm{rel}}$$

where O is the origin of the rotating link reference frame.

If the sliding path is radial along the rotating link and the pivot O is fixed, the radial and tangential components may be written:

The term:

$$a_{\text{Coriolis}}=2\omega v_{\text{rel}}$$

is the magnitude of the Coriolis acceleration for the planar case where the relative sliding velocity is perpendicular to the angular-velocity vector. Its vector form is:

$${\mathbf{a}}_{\text{Coriolis}}=2\mathbf{\omega }\times {\mathbf{v}}_{\text{rel}}$$

Coriolis acceleration occurs whenever there is relative motion along a member or guide that is itself rotating. Typical mechanism examples include a block sliding along a rotating link and a link sliding through a swivelling block.

---

Related RoyMech References

• Mechanism Help and Selection Guide
• Mechanical Mechanisms
• Cam and Follower Mechanisms
• Vectors
• Statics
• Kinematics
• Kinetics and Dynamics
• Simple Machines
• Component Loading
• Basic Strain Theory
• Failure Modes
• Buckling Notes
• Mohr's Circle
• Thick Cylinders
• Rotating Discs and Cylinders
• Torsion
• Stress-Strain Curve
• Loaded Flat Plates

External Linkage References

• Linkage Mechanisms – graphical examples of different linkage mechanisms.
• Mechanical Linkage – Wikipedia – general reference material on mechanical linkages.

�