magnitudes are used in timing charts. When used with two mechanisms, timing charts assume constant acceleration. This assumption produces polynomial equations for velocity as a function of time. Constant acceleration allows for the velocity vs. time graph to appear as straight lines, thus designating a relationship between displacement (ΔR), maximum velocity (vpeak), acceleration (a), and time(Δt). The following equations show this. Given the displacement and time, both the maximum velocity and acceleration of each mechanism in a given pair can be calculated. Slider-crank linkage A slider-crank linkage is a four-bar linkage with three revolute joints and one prismatic, or sliding, joint. The rotation of the crank drives the linear movement the slider, or the expansion of gases against a sliding piston in a cylinder can drive the rotation of the crank. There are two types of slider-cranks: in-line and offset. In-line An in-line slider-crank has its slider positioned so the line of travel of the hinged joint of the slider passes through the base joint of the crank. This creates a symmetric slider movement back and forth as the crank rotates. Offset If the line of travel of the hinged joint of the slider does not pass through the base pivot of the crank, the slider movement is not symmetric. It moves faster in one direction than the other. This is called a quick-return mechanism. Spherical and spatial four-bar linkages If the linkage has four hinged joints with axes angled to intersect in a single point, then the links move on concentric spheres and the assembly is called a spherical four-bar linkage. The input-output equations of a spherical four-bar linkage can be applied to spatial four-bar linkages when the variables are replaced by dual numbers. Bennett's linkage is a spatial four-bar linkage with hinged joints that have their axes angled in a particular way that makes the system movable. Examples Other Linkages and Mechanisms Chebyshev linkage Chebyshev lambda linkage Evans "Grasshopper" linkage Hoecken linkage Horse-head linkage Pantograph Roberts linkage Valve gear Watt's linkage Applications Bicycle suspension Biological linkages Double wishbone suspension Door closer Foldable steps and foldable chairs Foot operated machines (grindstone, lathe, sewing machine, treadle, etc.) Gear shifter Glider (furniture) Oscillating fan Pumpjack Step-on trash can Windshield wiper Simulations See also Burmester's theory Cognate linkage Five-bar linkage Kinematic synthesis Linkage (mechanical) Pumpjack Six-bar linkage Slider-crank linkage Spherical trigonometry Straight line mechanism (Approximate straight lines are primarily four-bar linkages) Universal