Analysis and synthesis of a controllable crank-slider mechanism with parallel springs for frame saws

1. Introduction

The effective processing of wood and wood-based materials relies on foundational cutting technologies [1]. In frame sawing operations, success is interconnected with several factors, including insights from tool condition monitoring [2], the wear performance of cutting tools [3], the overall parametric reliability of the machine tool [4], and the multi-objective optimization of components like the saw blade module [5] and general sawing parameters [6]. Furthermore, comparative studies on material properties, such as fracture toughness, are critical for understanding process limitations [7].

The main drive of frame saws, typically a crank-slider mechanism, inherently generates significant dynamic forces due to the reciprocating motion of large masses. The field of machine theory offers extensive knowledge for mitigating these issues. This includes established taxonomies for mechanism balancing [8], matrix methods for force analysis [9], techniques for managing kinetic energy fluctuations [10], and methods to reduce input torque and joint reactions in high-speed systems [15]. Specific research has focused on controllable crank mechanisms to manage oscillations [11, 12] and the potential utilization of mechanical resonance to either enhance or reduce dynamic loads [13, 14].

Beyond conventional cranks, alternative kinematic structures are explored for dynamic compensation. Research into planetary gear systems covers optimization design [16], detailed kinetostatic and force analysis [17, 18], and dynamic characteristics in demanding applications, including their use as inertial vibration exciters [19, 20, 21]. Similarly, cam mechanisms are frequently employed for precise motion control. Studies demonstrate their use for balancing and compensating drive torque [22, 24, 25], with design considerations for robustness [23], the rotational balancing of the cams themselves [26, 27], and profile optimization using advanced computational techniques [28, 29]. Synthesis methods are also developed for specific applications, such as specialized groove cam Geneva mechanisms [30].

The primary limitation of existing frame saws remains the large, unbalanced inertia load from the saw frame, which degrades cut quality, causes uneven tool wear, and restricts productivity by limiting operating speed. Increasing speed is impractical as the dynamic loads rise exponentially, necessitating massive foundations and powerful motors. This research aims to synthesize a controllable crank-slider mechanism with parallel springs where the inertia and spring forces are fully balanced throughout the entire stroke. The scientific novelty lies in synthesizing the drive mechanism based on a predetermined law of slider motion, using a target acceleration profile to overcome the inherent dynamic limitations of conventional designs.

2. Research methodology

2.1. Synthesis of the variable crank length based on a prescribed law of motion for the slider

The structural diagram of the mechanism under study is shown in Fig. 1. We denote the initial crank length in the extreme right position of the slider as $l_1=l_{O{A}_0}$, the variable crank length as $r=l_{O{A}_{11}}$, the initial connecting rod length as $l_2=l_{AB}$, the eccentricity as $e=l_{OE}$, the slider coordinates in the $xOy$ coordinate system as $x_B$, $y_B$, and the slider stroke as $S$. In non-dimensional (invariant) form, relative to the slider stroke $S$ (often normalized to $S=1$), we have: ${\lambda }_1=l_1/S\equiv l_1$, ${\lambda }_r=r/S\equiv r$, ${\lambda }_2=l_2/S\equiv l_2$, ${\lambda }_3=e\cdot \mathit{sgn}(y_B)/S\equiv e\cdot \mathit{sgn}(y_B)$, ${\lambda }_x=x_B/S\equiv x_B$, ${\lambda }_S=S/S\equiv 1$. We define the law of motion using the non-dimensional (invariant) displacement $a_k$ for the slider motion towards the crank rotation center, and $a_k^{\text{'}}$ for the motion away from the center. The normalization of the mechanism’s geometric dimensions relative to the stroke $S$ is chosen deliberately, since in the kinematic synthesis of crank-slider mechanisms, $S$, $e$, and the maximal dimension ratio, e.g. ${\lambda \mathrm{m}\mathrm{a}\mathrm{x}\mathrm{m}\mathrm{a}\mathrm{x}}_{x_{\mathrm{m}\mathrm{a}\mathrm{x}}}$, are typically specified. Using these parameters, the initial lengths of the crank and connecting rod will be calculated using the following relationships:

1

${\lambda }_1=\mathrm{0,5}\left[\sqrt{\left({\lambda }_{x_{\mathrm{m}\mathrm{a}\mathrm{x}}}^2+{\lambda }_3^2\right)}-\sqrt{{\left({\lambda }_{x_{\mathrm{m}\mathrm{a}\mathrm{x}}}-1\right)}^2+{\lambda }_3^2}\right]\equiv l_1;$
${\lambda }_2=\mathrm{0,5}\left[\sqrt{\left({\lambda }_{x_{\mathrm{m}\mathrm{a}\mathrm{x}}}^2+{\lambda }_3^2\right)}+\sqrt{{\left({\lambda }_{x_{\mathrm{m}\mathrm{a}\mathrm{x}}}-1\right)}^2+{\lambda }_3^2}\right]\equiv l_2.$

We can synthesize the variable crank length by solving the following mathematical problem: determining the intersection points between a line passing through the crank rotation center (O) and a circle of radius ${\lambda }_2$ (the length of the connecting rod AB) centered at point B (Fig. 1).

Let us represent the line in parametric form and write a system of equations for the coordinates of the intersection points. For clarity, all derivations will be carried out using dimensional quantities. Then, the system of equations is as follows:

2

$\left\{\begin{array}{c}x=r\cos ({\phi }_1),\\ y=r\sin ({\phi }_1),\\ {\left(x-x_B\right)}^2+{\left(y-y_B\right)}^2=l_2^2,\end{array}\right.$

where $x$, $y$ are the coordinates of the intersection points $A_{11}$ and $A_{12}$; $r=l_{OA}$ is the variable crank length.

Fig. 1Calculation scheme for determining the variable crank length based on the prescribed non-dimensional (invariant) slider displacement: 1 – crank; 2 – connecting rod; 3 – slider; 4 – sliding block

Substituting the expressions for $x$ and $y$ into the third equation of the Eq. (2) and performing straightforward transformations yields a quadratic equation for the variable crank length:

3

$r^2+pr+q=0,$

where $p=-2\left[x_B\cos \left({\phi }_1\right)+y_B\sin \left({\phi }_1\right)\right]$; $q=x_B^2+y_B^2-l_2^2$; ${\phi }_1=0...{\phi }_{1\mathrm{\Sigma }}$ and ${\phi }_1=0...{\phi }_{1\mathrm{\Sigma }}^{\text{'}}$ are the crank rotation angles for slider displacements of the same sign.

The roots of this equation are as follows:

4

$l_{O{A}_{11}}\equiv r_1=x_B\cos ({\phi }_1)+y_B\sin ({\phi }_1)-\sqrt{l_2^2-{\left[x_B\sin ({\phi }_1)-y_B\cos ({\phi }_1)\right]}^2},$
$l_{O{A}_{12}}\equiv r_2=x_B\cos ({\phi }_1)+y_B\sin ({\phi }_1)+\sqrt{l_2^2-{\left[x_B\sin ({\phi }_1)-y_B\cos ({\phi }_1)\right]}^2}.$

As the crank rotation angle ${\phi }_1$ increases, the distance between points $A_{11}$ and $A_{12}$ decreases, and at some angle ${\phi }_1={\phi }_{1p}$, these points coincide at point $A_{1p}$, and the line becomes tangent to the circle. This tangency condition determines the necessary length of the connecting rod $A_{1p}{B}_{1p}$ that ensures the slider moves according to the prescribed law for the chosen geometric dimensions of the mechanism. Let us denote this length as $l_{2s}=l_{A_{1p}{B}_{1p}}$ and determine it from the condition $r_2-r_1=0$ or $p^2-4q=0$. In the expanded form, this condition is as follows:

5

$z\equiv l_2^2-{\left[x_B\sin ({\phi }_1)-y_B\cos ({\phi }_1)\right]}^2=0.$

It is most practical to determine the value of the angle ${\phi }_1={\phi }_{1p}$, for which the condition defining the minimum of the function $z$ occurs, using numerical methods, since an analytical determination would still necessitate solving a nonlinear transcendental equation numerically.

Let us plot the function $z=f\left({\phi }_1\right)$ and determine the angle ${\phi }_1={\phi }_{1p}$ at which the function $z$ attains its minimum. Next, using Eq. (5), let us calculate the synthesized value of the connecting rod length:

6

$l_{2s}=x_B\left({\phi }_{1p}\right)\sin ({\phi }_{1p})-y_B\cos ({\phi }_{1p})$

Analysis of the obtained results shows that of the two real roots Eq. (4), for ${\phi }_1<{\phi }_{1p}$, the first root $r_1$ is valid, while for ${\phi }_1>{\phi }_{1p}$, the second root $r_2$ is valid. When the crank and connecting rod are mutually perpendicular, the crank radius is calculated using the obvious expression:

7

$r_{1p}=x_B\left({\phi }_{1p}\right)\cos ({\phi }_{1p}-{\alpha }_{1p})+y_B\sin ({\phi }_{1p}-{\alpha }_{1p}),$

where ${\alpha }_{1p}=\mathrm{atan}\left[e\cdot \mathrm{sgn}(y_B)/x_B({\phi }_{1p})\right]$.

Using the sign function, we obtain an expression for calculating the variable crank length that is valid for any crank rotation angle:

8

$r=x_B\cos ({\phi }_1)+y_B\sin ({\phi }_1)-\mathrm{sgn}({\phi }_{1p}-{\phi }_1)\sqrt{l_{2s}^2-{\left[x_B\sin ({\phi }_1)-y_B\cos ({\phi }_1)\right]}^2},$

where $x_B={xk}_{\mathrm{m}\mathrm{a}\mathrm{x}}$, $y_B=e\cdot \mathrm{sgn}(y_B)$ for ${\phi }_1<{\phi }_{1p}$; $x_B={xk}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ for ${\phi }_1>{\phi }_{1p}$; $x_B=x_B\left({\phi }_{1p}\right)$ for ${\phi }_1={\phi }_{1p}$; $a_k$ is the required non-dimensional (invariant) law of motion of the slider.

In the non-dimensional (invariant) form, the non-dimensional crank length is determined as follows:

9

${\lambda }_r={\lambda }_x\cos \left(k{\phi }_{\mathrm{\Sigma }}\right)+{\lambda }_3\sin \left(k{\phi }_{\mathrm{\Sigma }}\right)-\mathrm{sgn}({\phi }_{1p}-{\phi }_1)\sqrt{{\lambda }_{2s}^2-{\left[{\lambda }_x\sin \left(k{\phi }_{\mathrm{\Sigma }}\right)-{\lambda }_3\cos \left(k{\phi }_{\mathrm{\Sigma }}\right)\right]}^2},$

where $k=0...1$ is a new non-dimensional variable; ${\lambda }_{2s}=l_{2s}/S$; ${\phi }_{\mathrm{\Sigma }}={\phi }_{1\mathrm{\Sigma }}$ is the crank angle range during the slider displacement towards the crank rotation center and at ${\lambda }_x=\frac{x_B}{S}={\lambda }_{x\mathrm{m}\mathrm{a}\mathrm{x}}-a_k$; ${\phi }_{\mathrm{\Sigma }}={\phi }_{1\mathrm{\Sigma }}^{\text{'}}$ is the crank angle range during the displacement away from the rotation center and at ${\lambda }_x=x_B/S={\lambda }_{x\mathrm{m}\mathrm{a}\mathrm{x}}-1+a_k$.

The crank angle range is calculated using the following expressions:

10

${\phi }_{1\mathrm{\Sigma }}=\pi -\mathrm{sgn}({\omega }_1)\cdot \mathrm{\Delta }\phi , {\phi }_{1\mathrm{\Sigma }}^{\text{'}}=\pi +\mathrm{sgn}({\omega }_1)\cdot \mathrm{\Delta }\phi ,$
$\mathrm{\Delta }\phi =\mathrm{acos}\left(\frac{{\lambda }_3^2+{\lambda }_{x\mathrm{m}\mathrm{a}\mathrm{x}}({\lambda }_{x\mathrm{m}\mathrm{a}\mathrm{x}}-1)}{\sqrt{({\lambda }_3^2+{\lambda }_{x\mathrm{m}\mathrm{a}\mathrm{x}}^2)\left[{\lambda }_3^2+{\left({\lambda }_{x\mathrm{m}\mathrm{a}\mathrm{x}}-1\right)}^2\right]}}\right).$

Thus, the variable crank length, which provides the dynamic balancing of a crank-slider mechanism and ensures inertia-free motion of a slider (saw frame), can be calculated using Eqs. (8) or (9). Although the slider experiences variable velocity and acceleration, the inertia forces are fully balanced by the elastic forces of the parallel springs, which have the same functional dependence on displacement.

2.2. Synthesis of the variable crank length in the combined crank-slider mechanism with parallel spring placement

With parallelly positioned springs (Fig. 2), the elastic force acting on the slider varies according to a linear law and is $F_c=2c{x}_{def}$, where $c$ is the spring stiffness, $x_{def}=S\left(0.5-a_k\right)$ is the spring deformation, and $a_k$ is the non-dimensional slider displacement. To balance the slider, the slider’s law of motion must follow a cosine law, whose non-dimensional form is $a_k=0.5\left[1-\cos (\pi k)\right]$. In this case, the relationship between the slider’s acceleration and its displacement is also linear. Indeed, the non-dimensional acceleration is $c_k=d^2{a}_k/d{k}^2=0.5{\pi }^2\cos (\pi k)$ and $c_k={\pi }^2\left(0.5-a_k\right)$, which expresses a linear relationship. The inertia force of the slider is calculated using the obvious expression $F_{in}=c_k\frac{mS}{T^2}$, where $m$ is the unbalanced mass of the slider, and $T$ is the time for unidirectional slider displacement. For slider balancing, the inertia forces and the elastic forces of the springs must be equal in magnitude. Then, from the equality of these forces at the extreme position (if $k=0\to a_k=0$, $c_k={\pi }^2/2$), let us calculate the required spring stiffness for balancing the slider with parallel springs:

11

$c_{er.par}={{\lambda }_c}_{par}\cdot m{\omega }_1^2,$

where ${{\lambda }_c}_{par}={\pi }^2/\left(2{\phi }_{1\mathrm{\Sigma }}^2\right)$ is the non-dimensional (invariant) stiffness for parallel springs.

Fig. 2Structural diagram of a combined crank-slider mechanism with parallel springs: 1 – crank; 2 – connecting rod; 3 – slider; 4 – sliding block

The cosine periodic law of motion of the considered crank-slider mechanism with parallel springs is shown in Fig. 2. Next, using Eqs. (6) or (7), one can calculate the variable crank length (the theoretical radius vector of the fixed cam profile).

3. Results and discussion

3.1. Results of numerical modeling

For the numerical analysis, the following dimensions of a frame saw were used: saw frame stroke $S=$ 0.6 m, connecting rod length $l_2=$ 2 m, and eccentricity $e=$ 0 m. The number of spring pairs is set to $n_{sp}=1$ and the coefficient $k_b=1$.

Fig. 3 shows the required non-dimensional (invariant) kinematic characteristics for the slider that ensure it is balanced. In the case of parallelly positioned springs, these are the invariants of the cosine periodic law of motion. The maximum acceleration is characterized by the peak acceleration constant $C={\pi }^2/2=\text{4.935}$, and the maximum velocity – by the peak velocity constant $B=\pi /2=$ 1.57.

Fig. 4 shows the theoretical cam profile (variable crank length) and its pressure angles for the mechanism with parallelly positioned springs. In the synthesized mechanism, the connecting rod length also changed from ${\lambda }_2=\text{3.3333}$ to ${\lambda }_{2s}=\text{3.3698}$. The synthesized profile is very similar to a circle whose center is offset to the left (Fig. 4(a)). Calculations showed that when the profile is shifted to the right by 0.03747, the horizontal diameter of the circle would $d_{hor}=\text{1}$, and the vertical diameter $d_{ver}=0.\text{9896}$. Here, the average diameter differs by 0.5% from a circle with diameter $d=2{\lambda }_1=\text{1}$. This fact is a positive aspect, since the manufacturing of the cam could possibly be simplified to constructing a circular disc that rotates about an axis offset by 0.03747 from its geometric axis. However, this can only be confirmed after conducting an analysis of such a mechanism. The pressure angles of the synthesized profile do not exceed $\nu <$ 4.2° (Fig. 4(b)) and are much smaller than the permissible angle for cam mechanisms with translating followers ($\left[\nu \right]\approx$40°...50°).

Fig. 3Required non-dimensional kinematic characteristics of the slider for its balancing: ak – displacement; bk – velocity; ck – acceleration

Fig. 4Characteristics of the fixed cam for the mechanism with parallel springs

a) Theoretical cam profile

b) Cam pressure angles

4. Discussion of the results obtained

The results demonstrate the successful synthesis of the proposed mechanism with parallel springs, achieving the primary goal of deriving such a configuration that theoretically eliminates net dynamic forces on the slider. It was shown that a simpler cosine law of motion $\left(a_k=0.5\left[1-\cos (\pi k)\right]\right)$ achieves a balance due to the linear nature of both the resulting inertia force $\left(F_{in}\propto 0.5-a_k\right)$ and the parallel spring force $\left(F_{с}\propto 0.5-a_k\right)$. Based on these synthesized motion laws, the required variable crank length functions $r\left({\phi }_1\right)$ (equivalent to the theoretical cam profile ${\lambda }_r$) were determined using the derived analytical Eq. (8) or (9). Furthermore, the necessary geometric parameters for implementation, specifically the minimum required connecting rod length ($l_{2s}$ or ${\lambda }_{2s}$) and the required spring stiffness ($c$ or ${\lambda }_c$), were calculated based on the tangency condition and force balance at extreme positions, respectively. Crucially, the analysis of the pressure angles associated with the synthesized cam profile indicated that they remain within permissible limits (less than 4.5°), suggesting the feasibility of such a design from a cam-follower interaction perspective.

The proposed synthesis methodology offers a distinct alternative to conventional approaches for mitigating inertial forces in reciprocating machines. While methods like adding counterweights [8-10], utilizing specifically designed balancing cams [24-27], employing planetary mechanisms [16-19], or leveraging resonance exist [13, 14], the approach here focuses on tailoring the fundamental kinematics of the main drive itself. By defining a desired output motion law $\left(a_k\right)$ that results in force cancellation and then synthesizing the input mechanism (variable crank length $r\left({\phi }_1\right)$ or cam profile ${\lambda }_r$) required to produce it, this work embodies a direct kinematic synthesis approach to dynamic balancing. This is particularly relevant for frame saws where the large reciprocating mass of the saw frame is the primary source of vibration and dynamic load. The variable-length crank mechanism, actuated by a cam, functions as a form of programmable crank, allowing the generation of complex motion profiles beyond what a standard crank-slider can produce, akin to capabilities found in more complex cam-linkage or controllable crank systems.

The practical implementation hinges on the manufacturability of the required variable crank length mechanism, primarily the fixed cam profile. For the mechanism with parallel springs, the synthesized cam profile was found to be nearly circular, differing from a true circle by less than 0.5 % in average diameter (based on calculated horizontal and vertical extents). This suggests the possibility of realizing this mechanism using a simple eccentric circular cam, allowing for a significant simplification compared to manufacturing a complex non-circular profile. This finding substantially enhances the practical potential of the parallel spring configuration, although further analysis of the dynamic performance with an actual eccentric circular cam is warranted.

It should be emphasized that the calculated stiffness of the springs was derived exclusively from the condition of equilibrium between the inertia forces of the reciprocating slider and the restoring forces of the parallel springs. This approach ensures that the primary source of dynamic loading – the inertial action of the saw frame – is effectively compensated. Cutting forces, although present in real sawing processes, are significantly smaller in magnitude compared to the inertia forces at the operating speeds under consideration [1, 6, 7]. Therefore, their influence on the determination of spring stiffness was neglected in the present study. Nevertheless, for a more comprehensive dynamic model of the mechanism, the effect of cutting forces on the overall balance will be analyzed in future research.

5. Conclusions

This paper successfully addressed the objective of synthesizing dynamically balanced drive mechanisms for frame saws by employing a novel crank-slider mechanism with a cam-actuated variable-length crank. For the mechanism with parallel springs, the balancing was achieved using a cosine law of motion $\left(a_k=0.5\left[1-\cos (\pi k)\right]\right)$, which linearly relates slider acceleration to displacement, matching the linear characteristic of the parallel springs. The required theoretical cam profile, calculated via Eq. (9), was found to be nearly circular, with horizontal and vertical diameters differing minimally ($d_{hor}=\text{1}$, $d_{ver}=0.\text{9896}$ in non-dimensional terms) and the average diameter differing by less than 0.5 % from a perfect circle $\left(d=2{\lambda }_1=1\right)$. This suggests a high potential for simplification using an eccentric circular cam. The required non-dimensional connecting rod length was ${\lambda }_{2s}=$ 3.3698, and the pressure angles were very low (below 4.2°).

The proposed synthesis methodology allows for the design of frame saw mechanisms where the primary inertia loads are theoretically eliminated, achieved through specific slider motion laws generated by a variable-length crank mechanism. The parallel spring configuration offers the significant practical advantage of potentially being realized with a simple eccentric circular cam. These findings pave the way for frame saws capable of higher operating speeds, reduced vibration, lighter construction, and improved overall performance.

Future research will focus on detailed dynamic simulations of the synthesized mechanisms, incorporating friction, component elasticity, and cutting loads, followed by the development and experimental testing of physical prototypes to validate the theoretical balancing effectiveness and performance benefits in real-world conditions. Analysis of the simplified eccentric circular cam implementation is also a key next step.