A high-precision mechanical crankshaft assembly with a piston, connecting rod, and labeled balancing masses (reciprocating $m_{rec}$ and rotating $m_{rot}$) in a bright engineering workshop, illustrating the first-order inertial forces and asymmetric crankshaft balancing on a $X-Y-Z$ axis grid.

Understanding Motorcycle Engine Vibrations: First-Order Inertial Forces and Asymmetric Crankshaft Balancing (Part 1)

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  • Published on: July 14, 2026
  • Author: Johnny Liu, Chief Executive Officer at Dowway Vehicle
  • Expertise: 20+ years in motorcycle powertrain engineering, high-performance engine dynamics, and chassis integration.
  • Estimated Reading Time: 12 minutes
  • Target Audience: Powertrain Design Engineers, Motorcycle Enthusiasts, Mechanical Engineering Researchers.

Author’s Note

“When we build motorcycles, we do not try to completely destroy vibrations. Instead, we learn to shape them. Balancing a single-cylinder or parallel-twin engine is always about compromise. In this post, we will break down the basic physics of First-Order (Primary) Inertial Forces, see how they turn into the Inertial Force Ellipse, and look at the math we use to balance Asymmetric Crankshaft setups.”

— Johnny Liu, CEO at Dowway Vehicle

1. The Physics of Engine Vibration

Every internal combustion engine has parts that shake back and forth or spin around. When the piston moves up and down, it turns the crankshaft. This motion creates forces that make the motorcycle shake.

To understand these shakes, we split the forces into two main groups:

  1. Gas Forces: The push from burning fuel inside the cylinder.
  2. Inertial Forces: The forces created by heavy metal parts speeding up and slowing down.

Here, we will look only at First-Order (Primary) Inertial Forces. These forces happen at the exact same speed as the spinning crankshaft (1x engine speed, shown as $\omega$).

2. Reciprocating vs. Rotating Masses

Before doing any math, we must split the moving engine parts into two groups based on how they move.

                  [ Piston Assembly ]  <-- Reciprocating Mass (m_rec)
                         |
                         | (Small-end)
                   [ Connecting Rod ]  
                         | (Big-end)
                         |
                  [ Crankshaft Web ]   <-- Rotating Mass (m_rot)

A. Purely Reciprocating Mass ($m_{rec}$)

These parts move up and down in a straight line along the cylinder:

  • The piston, piston rings, and wrist pin.
  • The small end of the connecting rod.
  • About $1/3$ of the middle part (shank) of the connecting rod.

$$m_{rec} = m_{piston} + m_{pin} + m_{rings} + m_{rod\_small}$$

B. Purely Rotating Mass ($m_{rot}$)

These parts spin in a circle around the main centerline of the engine:

  • The crankpin.
  • The big end of the connecting rod and its bearings.
  • About $2/3$ of the connecting rod shank.
  • The counterweights on the crankshaft webs.

$$m_{rot} = m_{crankpin} + m_{rod\_big} + m_{webs}$$

What is First-Order Inertial Force?

First-order inertial force ($F_{I1}$) is the main shaking force in an engine that matches the speed of the crankshaft ($\omega$). The linear acceleration of the reciprocating mass ($m_{rec}$) creates this force along the cylinder path:$$F_{rec1} = m_{rec} \cdot r \cdot \omega^2 \cos\theta$$

Here, $r$ is the crank radius, $\omega$ is the crankshaft rotational speed, and $\theta$ is the crank angle from Top Dead Center (TDC).

3. The First-Order Inertial Force Ellipse

An unbalanced single-cylinder engine shakes hard along its cylinder line. To fix this, we put counterweights ($m_{cb}$) on the crankshaft opposite the crankpin.

If this weight balances a part ($C$) of the reciprocating mass, the spinning counterweight makes its own force:$$F_{cb} = C \cdot m_{rec} \cdot r \cdot \omega^2$$

We can split this counterweight force into vertical ($y$) and horizontal ($x$) parts, where the $y$-axis points along the cylinder:$$F_{cb, y} = -C \cdot m_{rec} \cdot r \cdot \omega^2 \cos\theta$$$$F_{cb, x} = -C \cdot m_{rec} \cdot r \cdot \omega^2 \sin\theta$$

The Combined Shaking Force

When we add the piston’s shaking force and the counterweight’s force together, we get the net forces pushing on the engine block:$$F_{net, y} = F_{rec1} + F_{cb, y} = (1 – C) \cdot m_{rec} \cdot r \cdot \omega^2 \cos\theta$$$$F_{net, x} = F_{cb, x} = -C \cdot m_{rec} \cdot r \cdot \omega^2 \sin\theta$$

These equations draw an ellipse in the $x-y$ plane. We call this shape the First-Order Inertial Force Ellipse.

How the Balance Factor ($C$) Changes the Ellipse Shape

Balance Factor ($C$)Vertical Force ($F_{net,y}$)Horizontal Force ($F_{net,x}$)Shape of the Force PathCommon Use Case
$C = 0$ (No Balance)$1.0 \cdot m_{rec} r \omega^2$$0$Straight Vertical LineUnusable; causes bad vertical shakes.
$0 < C < 0.5$LargeSmallVertical EllipseEngines mounted with soft vertical mounts.
$C = 0.5$ (50% Balance)$0.5 \cdot m_{rec} r \omega^2$$0.5 \cdot m_{rec} r \omega^2$Perfect CircleStandard starting point for single-cylinders.
$0.5 < C < 1.0$SmallLargeHorizontal EllipseEngines with horizontal cylinders.
$C = 1.0$ (100% Balance)$0$$1.0 \cdot m_{rec} r \omega^2$Straight Horizontal LineMoves all vertical shakes to horizontal shakes.

By changing $C$, we do not get rid of the shake. We just change its shape and direction. We match this force shape to the frame of the bike so the rider feels less vibration.

4. The Real World: Asymmetric Crankshaft Balancing

In school books, the left and right sides of a crankshaft look exactly the same. In real factories, they almost never do.

       [Left Crank Web]      [Conrod]      [Right Crank Web]
       +--------------+        | |         +--------------+
       |              |        | |         |              |
       |  Alternator/ |========| |=========|   Primary    |
       |  Magneto Side|   [Crankpin]       |  Drive Side  |
       |              |                    |              |
       +--------------+                    +--------------+
              |                                   |
        Shorter/Lighter                     Thicker/Heavier

Why the Sides are Different:

  1. Space Limits: The left side has to hold the alternator. The right side has to handle heavy drive gears and clutches. This makes the shafts different lengths and thicknesses.
  2. Unbalanced Weight: The physical shape and weight of the left and right metal webs are different.
  3. Rocking Motions (Couples): Since the left and right flywheels sit at different distances from the center line ($a_L \neq a_R$), any difference in their weight makes the engine twist and rock.

To stop the engine from twisting side-to-side, we must calculate the exact weights ($m_{cb,L}$ and $m_{cb,R}$) and angles ($\alpha_L$ and $\alpha_R$) for both sides separately.

5. Mathematical Modeling of Asymmetric Crankshaft Balancing

Let us set up a 3D grid where:

  • The Center ($O$) is where the cylinder line crosses the crankshaft spinning axis.
  • The $Y$-axis goes straight up along the cylinder.
  • The $X$-axis goes side-to-side (horizontal).
  • The $Z$-axis goes along the crankshaft.

Let:

  • $a_L$ and $a_R$ be the distances from the center ($z=0$) to the left and right crank webs.
  • $m_{rec}$ be the piston weight at the center line.
  • $m_{rot,pin}$ be the spinning weight of the crankpin and rod big end.
  • $m_{cb,L}$ and $m_{cb,R}$ be the counterweights on the left and right webs, at radii $r_L$ and $r_R$.
  • $\alpha_L$ and $\alpha_R$ be the angles of the weights relative to the direction opposite to the crankpin ($180^\circ$).
                      Y (Cylinder Centerline)
                       ^
                       |     Piston
                       |      [ ]
                       |       |
                       |       | Conrod
                       |       |
                       +-------*-------> Z (Crankshaft Axis)
                     / O      Crankpin
                    /
                   v
                  X (Horizontal Axis)

To keep the engine from shaking up, down, or twisting, we solve these equations together:

I. Balancing Forces in $Y$ and $X$ directions

The sum of all forces up, down, and sideways must match our target balance factor:$$\sum F_y = \left( m_{rec} \cdot r – m_{cb,L} \cdot r_L \cos\alpha_L – m_{cb,R} \cdot r_R \cos\alpha_R \right) \omega^2 \cos\theta + \left( m_{cb,L} \cdot r_L \sin\alpha_L – m_{cb,R} \cdot r_R \sin\alpha_R \right) \omega^2 \sin\theta = (1 – C) \cdot m_{rec} \cdot r \cdot \omega^2 \cos\theta$$$$\sum F_x = \left( m_{cb,L} \cdot r_L \cos\alpha_L + m_{cb,R} \cdot r_R \cos\alpha_R + m_{rot,pin} \cdot r \right) \omega^2 \sin\theta + \left( m_{cb,L} \cdot r_L \sin\alpha_L – m_{cb,R} \cdot r_R \sin\alpha_R \right) \omega^2 \cos\theta = -C \cdot m_{rec} \cdot r \cdot \omega^2 \sin\theta$$

These equations must work at every crank angle $\theta$. This means we can simplify them into two clean rules:$$m_{cb,L} \cdot r_L \cos\alpha_L + m_{cb,R} \cdot r_R \cos\alpha_R = C \cdot m_{rec} \cdot r + m_{rot,pin} \cdot r$$$$m_{cb,L} \cdot r_L \sin\alpha_L – m_{cb,R} \cdot r_R \sin\alpha_R = 0$$

II. Balancing Twisting Forces (Moments)

Because the left and right sides are at distances $a_L$ and $a_R$ from the center, any offset force will try to twist the engine. We keep the engine from twisting with these equations:$$\sum M_x = \left( m_{cb,L} \cdot r_L \cdot a_L \cos\alpha_L – m_{cb,R} \cdot r_R \cdot a_R \cos\alpha_R \right) \omega^2 \cos\theta = 0$$$$\sum M_y = \left( m_{cb,L} \cdot r_L \cdot a_L \sin\alpha_L + m_{cb,R} \cdot r_R \cdot a_R \sin\alpha_R \right) \omega^2 \sin\theta = 0$$

Solving these tells us the exact weight ($m_{cb,L}, m_{cb,R}$) and angle ($\alpha_L, \alpha_R$) we need for each side.

6. Practical Application at Dowway Vehicle

At Dowway Vehicle, we design the engine and the frame together. We shape the First-Order Inertial Force Ellipse to match the bike chassis:

  • Stiff Vertical Frames (Sportbikes): These frames are very stiff up and down, but have a bit of side-to-side give. We set the balance factor to $C \approx 0.60 \text{ to } 0.65$. This squishes the vertical shake and moves the energy to the sides, where the rubber engine mounts can absorb it.
  • Stiff Horizontal Frames (Cruisers): These frames are stiff front-to-back. We use $C \approx 0.45 \text{ to } 0.50$ here. This leaves a pleasant up-and-down pulse (the “heartbeat” of the engine) but keeps the bike from swaying.

7. Next Steps & Part 2 Preview

Handling first-order inertial forces is just the first step to a smooth engine. By understanding the Inertial Force Ellipse and calculation steps for Asymmetric Crankshafts, engineers can stop early engine wear and make the bike feel great to ride.

In Part 2, we will look closely at:

  • Second-Order (Secondary) Inertial Forces caused by the angle of the connecting rod.
  • Dual-Axis Balancer Shaft setups.
  • Crankshaft bending from cylinder firing pressure.

Have questions about balance factors or engine balancing formulas? Leave a comment below or contact our engineering team at Dowway Vehicle.

8. FAQs: Engine Inertial Forces & Balancing

Q1: Why can’t we completely balance a single-cylinder engine with just crankshaft weights?

A1: Because a single weight spins in a circle while the piston moves in a straight line. A spinning counterweight can cancel out the vertical movement of the piston, but as it swings to the side, it creates a brand-new horizontal shake of the exact same size. You only end up changing the direction of the shake, not getting rid of it.

Q2: Why do engineers use asymmetric crankshaft webs?

A2: To fit inside tight engine cases while keeping the engine from twisting. Different components on the left and right sides of the engine case require different shapes. Designing asymmetric counterweights helps balance out these shape differences, preventing the engine from rocking from side to side.

Q3: Does connecting rod length change first-order inertial forces?

A3: No, rod length does not affect first-order forces at all. First-order forces depend only on the piston weight ($m_{rec}$), crank stroke radius ($r$), and engine speed ($\omega$). Rod length only affects second-order forces, which we will talk about in Part 2.

© 2026 Dowway Vehicle Tech Blog. All Rights Reserved. For permissions to reprint, please contact engineering@dowwayvehicle.com.

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