Lecture 2: Kinematic Analysis of Mechanism

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Part 1 is organized around four connected topics: position analysis itself, graphical position analysis, algebraic position analysis, and the vector-loop method. The sequence matters because each method offers a different way to express the same geometric closure conditions.

The underlying aim is to move from geometric intuition toward a reusable analytical procedure. Graphical construction helps visualize configurations, algebraic coordinate methods solve directly for unknown point locations, and the vector-loop method provides a compact representation that can later be differentiated for velocity and acceleration analysis.

Position analysis is the starting point for nearly all later kinematic and dynamic calculations. The velocities of links and points are obtained by differentiating position with respect to time, and the accelerations follow from differentiating once again. Because dynamic forces depend on acceleration through Newton’s second law, reliable force and stress calculations depend on a correct position model first.

For a mechanism in continuous operation, the position problem is not solved at only one input angle. The configuration must be known throughout the motion cycle, so a useful analytical description should remain valid for a whole range of input positions rather than only a single drawn configuration.

Linkages can be analyzed graphically, analytically, through a complex-algebraic representation, through a vector-loop representation, or numerically. These are alternative ways of expressing the same constraint relationships rather than separate theories.

Graphical methods are often useful for intuition and quick construction, while algebraic and vector-loop methods are better suited to systematic derivation and computation. Numerical methods become especially useful when the mechanism is too complicated for a closed-form hand solution.

All position variables must be defined relative to a chosen reference frame, and the ground link is the usual reference in mechanism analysis.

A two-dimensional position vector may be written in polar form by its magnitude and angle or in Cartesian form by its $x$ and $y$ components, and the two forms are interchangeable.

A good coordinate choice does not change the physics, but it can simplify the equations substantially by aligning axes with known motions, fixed pivots, or slider directions.

Positions in a mechanism are described most clearly with vectors between named points and reference frames. Quantities such as the position of $D$ with respect to $O$, the position of $C$ with respect to $A$, and the relationship between local and global coordinates are all expressed naturally in vector form.

When a closed path is traced around a linkage, the vector sum must return to the starting point. That loop-closure idea is the basis of the later analytical method, because it converts the geometry of the linkage into equations that must hold at every admissible position.

Using the notation shown in the figure, the requested vector descriptions are:

1. Position of $D$ with respect to $O$: $\mathbf{R}_{DO}$
2. Position of $C$ with respect to $A$: $\mathbf{R}_{CA}$
3. Position of the local-origin point with respect to the global origin: $\mathbf{R}{O{\ell}O}$
4. Position of $C$ with respect to the local origin: $\mathbf{R}{CO{\ell}}$
5. For the closed loop passing through the points $O$, $A$, $B$, $C$, and back to $O$, the loop-closure condition is: $\mathbf{R}{AO}+\mathbf{R}{BA}+\mathbf{R}{CB}+\mathbf{R}{OC}=\mathbf{0}$.

An equivalent form is $\mathbf{R}{OA}+\mathbf{R}{AB}+\mathbf{R}{BC}+\mathbf{R}{CO}=\mathbf{0}$,

provided the subscripts are used consistently with the chosen direction of traversal.

Coordinate-system choice depends on the calculation being performed. A global coordinate system is useful for describing the mechanism as a whole, while a local coordinate system is often more convenient for describing a point fixed to a moving link.

The most effective reference frame is usually the one that makes the geometric constraints simplest. Aligning an axis with a slider direction, a grounded pivot, or a known line of motion often reduces both algebraic effort and the chance of sign errors.

If local coordinates $(R_x,R_y)$ are known and the local axes are rotated by an angle $\delta$ relative to the global axes, then a pure rotation of components gives

\[\begin{aligned} R_X &= R_x \cos \delta - R_y \sin \delta, \\ R_Y &= R_x \sin \delta + R_y \cos \delta \end{aligned}\]

This form is valid when the local and global origins coincide. More generally, if the local origin has global coordinates $(X_{O_{\ell}},Y_{O_{\ell}})$, then the full coordinate transformation is

\[\begin{aligned} R_X &= X_{O_{\ell}} + R_x \cos \delta - R_y \sin \delta, \\ R_Y &= Y_{O_{\ell}} + R_x \sin \delta + R_y \cos \delta \end{aligned}\]

This transformation is used repeatedly when points are fixed on moving links.

Displacement is the change in position of a point between an initial and a final configuration, measured in the selected reference frame.

Relative position instead describes one point with respect to another point at the same instant, and both are written with the same vector subtraction: $\mathbf{R}_{BA} = \mathbf{R}_B - \mathbf{R}_A.$

The physical interpretation depends on whether $A$ and $B$ represent successive positions of one body or simultaneous positions of two different bodies.

For pure translation, every point on the rigid body undergoes the same displacement. For corresponding points on the body, $\mathbf{R}{A^{\prime}A}=\mathbf{R}{B^{\prime}B}.$

The essential requirement is that the body does not rotate, so its orientation remains unchanged while all points undergo the same displacement.

This does not mean the motion must be along a straight line. Translation may be curvilinear as well as rectilinear. Rectilinear translation is only the special case in which the path is straight.

The notation order matters because reversing the subscripts reverses the vector direction and therefore changes the sign. Thus $\mathbf{R}{AA^{\prime}}=-\mathbf{R}{A^{\prime}A}.$

For pure rotation, points on the same body generally experience different displacements because each point lies at a different distance and direction from the center of rotation. One point or axis remains fixed, while all other points move on circular arcs about that center.

This means that the body does not have one single displacement vector that applies everywhere. A point close to the center moves through a short arc, while a point farther away moves through a longer arc for the same rotation angle. If the body rotates through an angle $\theta$, a point at radius $r$ moves through an arc length $s=r\theta.$

The displacement therefore depends on the position of the point within the body, not just on the fact that the body has rotated.

Displacement relations must therefore be written using the relative geometry between points rather than by assuming a common motion for the entire body. In translation, all points share the same displacement vector. In rotation, the motion of each point must be related to its distance and orientation relative to the center of rotation or to another reference point on the rigid body.

Complex planar motion is a combination of translation and rotation. In that case the total motion of a point is obtained by combining a rigid-body translation with a rotation about a moving point or frame, and the operations must be expressed in a consistent reference frame before they can be added correctly.

This is the standard motion state for coupler links in planar mechanisms. It is often described by Chasles’ theorem, which states that a general displacement of a rigid body can be represented as the translation of one point on the body together with a rotation about that point.

The order of those operations is generally important. Translation followed by rotation does not usually give the same final position as rotation followed by translation, because the second operation acts on a different intermediate configuration. Total displacement must therefore be built from a clearly defined sequence of motions.

Graphical position analysis determines the unknown configuration of a linkage by direct geometric construction. For a four-bar mechanism, known link lengths and an input angle can be used to draw arcs whose intersections locate the possible output positions.

These intersections usually correspond to the open and crossed configurations of the linkage. In the open configuration, the coupler and follower close the chain without crossing over the mechanism. In the crossed configuration, the chain closes on the opposite side, so the moving links cross relative to the open case. The link lengths and input angle are the same in both cases; what changes is the assembly mode of the mechanism.

The method is conceptually simple because the geometry is visible, but each input position must be redrawn independently. It is therefore useful for visualization and checking, but it is inefficient when a complete motion history is needed.

The algebraic method replaces the drawing process with coordinate equations.

Point $A$ is found from

\[\begin{aligned} A_X &= a \cos \theta_2, \\ A_Y &= a \sin \theta_2 \end{aligned}\]

Point $B$ is then constrained by the two link-length relations

\[\begin{aligned} b^2 &= (B_X - A_X)^2 + (B_Y - A_Y)^2, \\ c^2 &= (d - B_X)^2 + B_Y^2 \end{aligned}\]

These are simply the circle constraints imposed by the coupler and rocker lengths. Solving them gives the unknown coordinates of $B$, from which the remaining link angles follow by trigonometry.

If the computed values of $B_X$ and $B_Y$ are imaginary, the linkage cannot close for the chosen input angle, so that configuration is infeasible. If the solution is real, the remaining link angles can be recovered from the coordinates, and multiple real solutions may correspond to different assembly modes of the same mechanism.

In practical terms, the two real solutions correspond to the open and crossed configurations. Feasibility is therefore not only a matter of link lengths but also of whether the chosen input angle is compatible with closure.

The vector-loop method expresses each link as a vector and enforces loop closure around the mechanism. By writing those vectors in complex form and using Euler’s relation $e^{j\theta} = \cos \theta + j \sin \theta,$ the geometry of the four-bar can be converted into compact algebraic equations.

This representation is especially powerful because the same loop equation can later be differentiated directly to obtain velocity and acceleration relations without redefining the mechanism.

The fundamental loop equation for the four-bar is $\mathbf{R}A + \mathbf{R}{BA} - \mathbf{R}{BO_4} - \mathbf{R}{O_4} = 0.$ This states that if the loop is traversed from the ground pivot through the moving links and back to the fixed frame, the net displacement must be zero.

Any correct position solution must satisfy this equation exactly. The advantage of the vector form is that geometry, closure, and sign convention are all encoded in a single relation.

Replacing each vector with its complex polar form produces

\[a e^{j\theta_2} + b e^{j\theta_3} - c e^{j\theta_4} - d e^{j\theta_1} = 0.\]

Expanding the exponentials into sines and cosines converts the geometric closure problem into

\[a(\cos\theta_2 + j\sin\theta_2)+b(\cos\theta_3 + j\sin\theta_3)-c(\cos\theta_4 + j\sin\theta_4)-d(\cos\theta_1 + j\sin\theta_1)=0.\]

This is the point at which the vector equation becomes ready to split into scalar component equations. The real part represents horizontal closure, while the imaginary part represents vertical closure.

Separating the equation into real and imaginary parts gives the scalar closure relations

\[\begin{aligned} a\cos\theta_2 + b\cos\theta_3 - c\cos\theta_4 - d\cos\theta_1 &= 0, \\ a\sin\theta_2 + b\sin\theta_3 - c\sin\theta_4 - d\sin\theta_1 &= 0 \end{aligned}\]

If the ground link is aligned with the $x$-axis, then $\theta_1=0$, so the real equation contains $-d$ and the imaginary equation loses the ground-link term entirely.

The next step is to rearrange the two scalar equations into a form involving only one unknown angle at a time:

\[\begin{aligned} (2ab\cos\theta_2 - 2bd)\cos\theta_3 + 2ab\sin\theta_2\sin\theta_3 + (a^2+b^2+d^2-c^2-2ad\cos\theta_2) &= 0, \\ (2cd-2ac\cos\theta_2)\cos\theta_4 - 2ac\sin\theta_2\sin\theta_4 + (a^2-b^2+c^2+d^2-2ad\cos\theta_2) &= 0 \end{aligned}\]

These equations are still trigonometric, but each now contains only one unknown angle.

For the equation in $\theta_4$, it is convenient to define

\[\begin{aligned} A &= 2cd - 2ac\cos\theta_2, \\ B &= -2ac\sin\theta_2, \\ C &= a^2-b^2+c^2+d^2-2ad\cos\theta_2 \end{aligned}\]

so that the equation becomes $A\cos\theta_4 + B\sin\theta_4 + C = 0.$

This compact form isolates the dependence on $\theta_4$ and prepares the equation for a standard trigonometric substitution.

For the equation in $\theta_3$, define

\[\begin{aligned} D &= 2ab\cos\theta_2 - 2bd, \\ E &= 2ab\sin\theta_2, \\ F &= a^2+b^2+d^2-c^2-2ad\cos\theta_2 \end{aligned}\]

so that $D\cos\theta_3 + E\sin\theta_3 + F = 0.$

The structure is exactly the same as the previous case: one cosine term, one sine term, and one constant term.

The final analytical step is to apply a tangent half-angle substitution, such as

\[\begin{aligned} t &= \tan\left(\frac{\theta}{2}\right), \\ \cos\theta &= \frac{1-t^2}{1+t^2}, \\ \sin\theta &= \frac{2t}{1+t^2} \end{aligned}\]

Substituting this into an equation of the form $A\cos\theta + B\sin\theta + C = 0$ produces a quadratic in $t$, $(C-A)t^2 + 2Bt + (A+C) = 0,$

from which the two possible angle roots follow. In general, two real roots correspond to two admissible assembly configurations, while imaginary roots indicate infeasible closure.

The full derivation is worked through in the notes on Vector Loop Position Analysis.

The same vector-loop logic carries directly to the four-bar crank-slider once the slider position is treated as a linear unknown. Using the sign convention adopted here, a convenient closure statement is $\mathbf{R}A + \mathbf{R}{BA} - \mathbf{R}_{BO_2} = 0,$ which enforces closure between the crank, coupler, and slider-position vectors.

In complex form, with the same convention for $\theta_3$, this becomes $a e^{j\theta_2} - b e^{j\theta_3} - (d + jc) = 0,$ so the real and imaginary parts give

\[\begin{aligned} a\cos\theta_2 - b\cos\theta_3 - d &= 0, \\ a\sin\theta_2 - b\sin\theta_3 - c &= 0 \end{aligned}\]

The method is therefore unchanged in structure: write the loop equation, expand it, separate components, and solve for the unknown position variables.

For the crank-slider, the imaginary equation can be solved first to obtain the coupler-angle branches: $\sin\theta_3=\frac{a\sin\theta_2-c}{b}.$

This produces two assembly modes, one from the principal inverse-sine value and one from the supplementary angle, corresponding to the open and crossed configurations.

Once the appropriate value of $\theta_3$ has been chosen, the slider position follows from the real equation: $d = a\cos\theta_2 - b\cos\theta_3.$

This is the main transfer result for Week 2: the same vector-loop procedure used for the four-bar gives a direct position-analysis route for the crank-slider with one angular unknown and one linear unknown.

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Velocity can be analyzed by graphical, analytic, complex-algebraic, vector-loop, or numerical methods. These are different representations of the same kinematic constraints rather than unrelated theories.

For this course the vector-loop methods are the main focus because they give a direct route from solved positions to solved velocities and can be extended systematically into acceleration analysis.

Velocity analysis starts after the mechanism configuration has already been found. The objective is to determine the angular velocities of the links and the linear velocities of the points of interest in the mechanism.

These velocities are needed because acceleration analysis depends on them, and dynamic force calculations depend on acceleration. The topic therefore sits between position analysis and the later force and stress calculations.

For a point $P$ on a rotating link relative to point $A$, the position vector is written as $\mathbf{R}_{PA}=p e^{j\theta}.$

Differentiating with respect to time gives

\[\mathbf{V}_{PA}=\frac{d\mathbf{R}_{PA}}{dt}=j p \omega e^{j\theta}=p\omega(-\sin\theta+j\cos\theta)\]

The magnitude is therefore $p\omega$, and the direction is perpendicular to the radius vector. The required inputs are the distance $p$, the angular position $\theta$, and the angular velocity $\omega$.

When two points lie on the same rigid body, the appropriate relation is the velocity-difference equation $\mathbf{V}P=\mathbf{V}_A+\mathbf{V}{PA}.$

In that case $\mathbf{V}_{PA}$ is the rotational contribution generated by the rigid body motion between the two points.

When two points belong to different bodies, a single rigid-body velocity relation no longer applies. Their velocities must instead satisfy the constraint imposed by the joint or contact between the bodies. For example, a common pin joint must have the same instantaneous velocity when described from either link.

Velocity analysis for the four-bar begins from the same loop equation used in the position analysis, $\mathbf{R}A+\mathbf{R}{BA}-\mathbf{R}{BO_4}-\mathbf{R}{O_4}=0.$

The key advantage of the vector-loop method is that once the position equation is known, the velocity equation follows directly by differentiation.

Replacing the vectors with their complex polar forms gives $a e^{j\theta_2}+b e^{j\theta_3}-c e^{j\theta_4}-d e^{j\theta_1}=0.$

Differentiating with respect to time produces a velocity loop involving the angular rates of the moving links:

\[\begin{aligned} j a e^{j\theta_2}\frac{d\theta_2}{dt} &+j b e^{j\theta_3}\frac{d\theta_3}{dt} \\ &-j c e^{j\theta_4}\frac{d\theta_4}{dt} \\ &-j d e^{j\theta_1}\frac{d\theta_1}{dt}=0 \end{aligned}\]

The term containing $\theta_1$ comes from differentiating the ground-link vector. In the general form it is kept so that the differentiation is shown consistently for all four links.

For the usual four-bar reference frame, however, the ground link is fixed and aligned with the $x$-axis, so $\theta_1$ is constant and therefore $\frac{d\theta_1}{dt}=0.$

That makes the ground-link term vanish, leaving only the moving-link terms in the velocity equation.

With $\omega=d\theta/dt$ and Euler’s relation $e^{j\theta}=\cos\theta+j\sin\theta$, the differentiated loop becomes $j a \omega_2 e^{j\theta_2}+j b \omega_3 e^{j\theta_3}-j c \omega_4 e^{j\theta_4}=0.$

Expanding and separating into components gives the scalar equations

\[\begin{aligned} -a\omega_2\sin\theta_2-b\omega_3\sin\theta_3+c\omega_4\sin\theta_4 &= 0, \\ a\omega_2\cos\theta_2+b\omega_3\cos\theta_3-c\omega_4\cos\theta_4 &= 0 \end{aligned}\]

These two equations are then solved for the unknown angular velocities $\omega_3$ and $\omega_4$.

Solving the two scalar equations simultaneously gives

\[\begin{aligned} \omega_3 &= \omega_2\frac{a\sin(\theta_4-\theta_2)}{b\sin(\theta_3-\theta_4)}, \\ \omega_4 &= \omega_2\frac{a\sin(\theta_2-\theta_3)}{c\sin(\theta_4-\theta_3)} \end{aligned}\]

These expressions are valid only after the position analysis has supplied the current values of $\theta_3$ and $\theta_4$.

Once the angular velocities are known, point velocities follow directly from

\[\begin{aligned} \mathbf{V}_A &= a\omega_2(-\sin\theta_2+j\cos\theta_2), \\ \mathbf{V}_{BA} &= b\omega_3(-\sin\theta_3+j\cos\theta_3), \\ \mathbf{V}_B &= c\omega_4(-\sin\theta_4+j\cos\theta_4) \end{aligned}\]

The crank-slider transfer case begins from the corresponding loop equation $\mathbf{R}A+\mathbf{R}{BA}-\mathbf{R}_{BO_2}=0.$ The analytical logic is unchanged: start from the solved position state, differentiate with respect to time, and then separate the result into scalar equations for the unknown instantaneous velocities.

For the four-bar crank-slider, the velocity relations are

\[\begin{aligned} \omega_3 &= \frac{a\cos\theta_2}{b\cos\theta_3}\omega_2, \\ \dot d &= -a\omega_2\sin\theta_2+b\omega_3\sin\theta_3 \end{aligned}\]

The first equation gives the coupler angular velocity, while the second gives the slider speed. The transfer from the four-bar is therefore direct: the same differentiated loop-closure logic is used, but one unknown is now linear rather than angular.

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Acceleration analysis follows directly from velocity analysis. Once the velocity relations of the mechanism are known, a further differentiation yields the linear and angular accelerations required for dynamic force calculations and inertia-based stress estimation.

The same vector-loop structure remains in place, but the resulting equations now contain both tangential terms proportional to angular acceleration and normal terms proportional to squared angular velocity.

Acceleration analysis matters because dynamic force depends directly on acceleration. If the accelerations of the links and points are known, then inertial loading can be estimated and used to evaluate the resulting forces and stresses in the mechanism components.

In rigid-body motion, acceleration is more informative than velocity because it separates naturally into geometric components that indicate whether a point is speeding up along its path or being pulled toward a center of curvature.

Angular acceleration is denoted by $\alpha$, while linear acceleration is denoted by $\mathbf{A}$.

Clear notation matters because rotational and translational effects appear simultaneously in the acceleration equations, and each term must be interpreted correctly as either tangential or normal motion.

For a point at radius $r$ on a rotating link, the tangential acceleration has magnitude $r\alpha$ and the normal, or centripetal, acceleration has magnitude $r\omega^2$ directed toward the center of rotation.

The conventional four-bar linkage remains the reference mechanism for the derivation. The same geometry used in the position and velocity analysis is carried forward so that the new unknowns are accelerations rather than positions or angular velocities.

Starting from the velocity loop $j a \omega_2 e^{j\theta_2} + j b \omega_3 e^{j\theta_3} - j c \omega_4 e^{j\theta_4} = 0,$ differentiating once more with respect to time introduces both angular-acceleration terms and centripetal terms proportional to $\omega^2$. In complex form, differentiation of $j\omega e^{j\theta}$ generates both a $j\alpha e^{j\theta}$ term and a $-\omega^2 e^{j\theta}$ term.

That is why acceleration analysis contains both tangential and normal components.

After regrouping, each link contribution naturally separates into a tangential part proportional to $\alpha$ and a normal part proportional to $\omega^2$. The acceleration loop can then be written as $\mathbf{A}A + \mathbf{A}{BA} - \mathbf{A}_B = 0.$

This is the acceleration version of loop closure for the four-bar.

Each vector in this equation represents the sum of a component tangent to the motion and a component directed inward toward the center of curvature.

The immediate objective is to find the unknown angular accelerations $\alpha_3$ and $\alpha_4$. The known quantities are the input angular acceleration $\alpha_2$, the link lengths, the current link angles, and the angular velocities already found from the previous stage of the analysis.

This dependency shows that acceleration analysis cannot be carried out independently. It requires the results of both the position and velocity analyses.

The solution strategy is the same as before: expand the complex equation, separate it into real and imaginary components, and then solve the resulting pair of scalar equations simultaneously. This repetition is one of the main strengths of the vector-loop method, because the same structure survives across position, velocity, and acceleration analysis.

The acceleration loop is split into real and imaginary component equations. These scalar relations combine the tangential terms in $\alpha_2$, $\alpha_3$, and $\alpha_4$ with the normal terms in $\omega_2^2$, $\omega_3^2$, and $\omega_4^2$, producing the two equations needed to solve for the unknown angular accelerations.

Once the current geometry and angular velocities are known, these equations are linear in $\alpha_3$ and $\alpha_4$.

Written explicitly, the two scalar acceleration equations are

\[\begin{aligned} -a\alpha_2\sin\theta_2-a\omega_2^2\cos\theta_2-b\alpha_3\sin\theta_3-b\omega_3^2\cos\theta_3+c\alpha_4\sin\theta_4+c\omega_4^2\cos\theta_4 &= 0, \\ a\alpha_2\cos\theta_2-a\omega_2^2\sin\theta_2+b\alpha_3\cos\theta_3-b\omega_3^2\sin\theta_3-c\alpha_4\cos\theta_4+c\omega_4^2\sin\theta_4 &= 0 \end{aligned}\]

These are solved simultaneously for $\alpha_3$ and $\alpha_4$.

Solving the two component equations simultaneously yields explicit expressions for $\alpha_3$ and $\alpha_4$. The formulas are longer than the velocity results, but the structure is the same: known geometry and known motion of the input link determine the unknown motion of the remaining links.

Once $\alpha_3$ and $\alpha_4$ are known, the accelerations of the important points follow directly from the complex rigid-body formulas. Each expression contains a tangential term proportional to $\alpha$ and a normal term proportional to $\omega^2$, which together describe the full acceleration of the point on the rotating link.

For the four-bar linkage, the principal point accelerations may be written as

\[\begin{aligned} \mathbf{A}_A &= a\alpha_2(-\sin\theta_2+j\cos\theta_2)-a\omega_2^2(\cos\theta_2+j\sin\theta_2), \\ \mathbf{A}_{BA} &= b\alpha_3(-\sin\theta_3+j\cos\theta_3)-b\omega_3^2(\cos\theta_3+j\sin\theta_3), \\ \mathbf{A}_B &= c\alpha_4(-\sin\theta_4+j\cos\theta_4)-c\omega_4^2(\cos\theta_4+j\sin\theta_4) \end{aligned}\]

The real parts give the $x$-components and the imaginary parts give the $y$-components of the corresponding acceleration vectors.

These point accelerations are the quantities needed when translating kinematic results into dynamic loading on pins, links, joints, and other connected components.

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The crank-slider uses the same loop-closure idea as the four-bar, but one end point is constrained to remain on the slider axis. The loop $\mathbf{R}A+\mathbf{R}{BA}-\mathbf{R}_{BO_2}=0$ enforces closure between the crank, coupler, and slider-position vectors.

Because point $B$ is restricted to the guide, its motion contains a translational coordinate along the slider axis rather than a second rotating link angle. That is the main geometric difference between the four-bar and the crank-slider formulations.

The crank-slider acceleration analysis follows the same sequence as the four-bar analysis. Position analysis supplies the current geometry, velocity analysis supplies $\omega_3$ and the slider speed, and one more differentiation yields the coupler angular acceleration and the slider acceleration.

The resulting acceleration relations are

\[\begin{aligned} \alpha_3 &= \frac{a\alpha_2\cos\theta_2-a\omega_2^2\sin\theta_2+b\omega_3^2\sin\theta_3}{b\cos\theta_3}, \\ \ddot d &= -a\alpha_2\sin\theta_2-a\omega_2^2\cos\theta_2+b\alpha_3\sin\theta_3+b\omega_3^2\cos\theta_3 \end{aligned}\]

The first equation gives the angular acceleration of the coupler, and the second gives the linear acceleration of the slider along its guide.

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Knowledge Check