Double Wishbone Suspension

This type of suspension is widely used in various vehicles, from road cars to competition vehicles.

The position of the spring and damper in the suspension system can be adapted to various configurations, meaning they can be placed between different elements of the system.

Initially, we will analyze the base kinematics without the spring and damper system. This approach allows us to study a simpler system, verifying and understanding its operation, and able to study the positions and orientation of the wheel under study. Later, we will develop complete systems with various spring/damper configurations by slightly increasing the complexity of the initially defined system.

Key Components

• Upper Control Arm (UCA): The upper wishbone or control arm connects the upper part of the wheel hub to the vehicle chassis. It helps control the vertical movement of the wheel and maintains the wheel’s alignment.

• Lower Control Arm (LCA): The lower wishbone or control arm connects the lower part of the wheel hub to the vehicle chassis. It works in conjunction with the upper control arm to manage the wheel’s movement and alignment.

• Knuckle or Upright: This component connects the upper and lower control arms to the wheel hub. It allows the wheel to pivot for steering and supports the wheel bearings.

• Tie Rod: In the case of front suspensions, the tie rod connects the steering rack to the wheel hub, allowing for steering input to be transmitted to the wheels.

• Spring and Damper: These components are crucial for absorbing shocks and maintaining ride comfort. The position of the spring and damper in the suspension system can be adapted to various configurations, meaning they can be placed between different elements of the system.

Configuration: Base

Note

For an interactive visualization of the suspension, view the CAD example representation: Double Whisbonebase: Base.

The base system of the Double Wishbone suspension is defined using 9 points, as presented in Points Defining the Base Wishbone Suspension. Of these, 5 points represent the vehicle chassis and are considered fixed, as the kinematic analysis is relative to these coordinates.

By convention, chassis points (fixed points) are named with letters (UCA FRONT, UCA REAR, LCA FRONT, LCA REAR), while mobile points are named with numbers (uca outer, lca outer, tierod outer, wheel center). This convention is used in all analyses. This standard definition uses Lca (lower control arm) and Uca (upper control arm).

The TIEROD INNER point refers to the steering system (or toe link in the case of rear suspension), which is considered fixed in the analysis. Therefore, the system has a single degree of freedom, requiring a total of 11 constraint equations. To account for the steering behavior, several analyses must be conducted by varying this point’s coordinates.

Points Defining the Base Wishbone Suspension

Points Name	Description	Type
wheel center	Center of the Wheel	mobile
UCA FRONT	Upper Control Arm Front	fixed
UCA REAR	Upper Control Arm Rear	fixed
uca outer	Upper Control Arm Outer	mobile
LCA FRONT	Lower Control Arm Front	fixed
LCA REAR	Lower Control Arm Rear	fixed
lca outer	Lower Control Arm Outer	mobile
TIEROD INNER	Inner Tie Rod	fixed
tierod outer	Outer Tie Rod	mobile

The 11 constraint equations necessary to define the system are presented in table Constraint Equations: Base Wishbone Suspension.

Note that all the constraint equations are based on the distance between two points, defined as the square root of the sum of the squares of the differences of the coordinates of the points. With this type of constraint equation, combining several types, it is possible to define the wishbones, tie rod, and all the components. The constraint equations will be represented as explained in Convention for Writing Proposals.

Constraint Equations: Base Wishbone Suspension

Equation	Part Definition	Initial Point	Final Point
1	Upper Wishbone	uca outer	UCA FRONT
2		uca outer	UCA REAR
3	Lower Wishbone	lca outer	LCA FRONT
4		lca outer	LCA REAR
5	Tie Rod	tierod outer	TIEROD INNER
6	Knuckle	tierod outer	uca outer
7		tierod outer	lca outer
8		lca outer	uca outer
9	Wheel Center	wheel center	uca outer
10		wheel center	lca outer
11		wheel center	tierod outer

Configuration: 1

Note

For an interactive visualization of the suspension, view the CAD example representation: Double Whisbone: Configuration 1.

In this configuration, the spring-damper assembly, is introduced. The lower point of the suspension (l spring mount) is connected to the lower wishbone. The other end (U SPRING MOUNT) is connected directly to the vehicle chassis.

The system is defined using a total of 11 points, of which 5 represent the vehicle chassis and are considered fixed. It can be observed that the definition of this configuration is identical to that used to define the base system, with the addition of the connection point between the suspension and the lower wishbone (l spring mount), as well as the point U SPRING MOUNT, which fixes the other end of the suspension to the vehicle chassis, which will not influence the kinematic analysis.

Therefore, the system has a total of 5 mobile points, resulting in a total of 15 natural coordinates.

Again, the system has a single degree of freedom. Thus, a total of 14 constraint equations will be necessary. Of these, 11 equations represent the base of the suspension system, and the remaining 3 define the position of the point l spring mount on the lower wishbone.

Points Defining the Wishbone Suspension Configuration 1

Points Name	Description	Type
wheel center	Center of the Wheel	mobile
UCA FRONT	Upper Control Arm Front	fixed
UCA REAR	Upper Control Arm Rear	fixed
uca outer	Upper Control Arm Outer	mobile
LCA FRONT	Lower Control Arm Front	fixed
LCA REAR	Lower Control Arm Rear	fixed
lca outer	Lower Control Arm Outer	mobile
TIEROD INNER	Inner Tie Rod	fixed
tierod outer	Outer Tie Rod	mobile
U SPRING MOUNT	Spring/damper Upper Mount	fixed
l spring mount	Spring/Damper Lower Mount	mobile

Constraint Equations: Configuration 1 Double Wishbone Suspension

Equation	Part Definition	Initial Point	Final Point
1	Upper Wishbone	uca outer	UCA FRONT
2		uca outer	UCA REAR
3	Lower Wishbone	lca outer	LCA FRONT
4		lca outer	LCA REAR
5	Tie Rod	tierod outer	TIEROD INNER
6	Knuckle	tierod outer	uca outer
7		tierod outer	lca outer
8		lca outer	uca outer
9	Wheel Center	wheel center	uca outer
10		wheel center	lca outer
11		wheel center	tierod outer
12	Point Suspension Lower Wishbone	l spring mount	LCA FRONT
13		l spring mount	LCA REAR
14		l spring mount	lca outer

Configuration: 2

Note

For an interactive visualization of the suspension, view the CAD example representation: Double Whisbone: Configuration 2.

In this configuration, the suspension is connected to a rocker that is actuated by a rod, which has its other end connected to the upper wishbone of the system.

The lower point of the suspension (l spring mount) is connected to the rocker. The other end (UPPER SPRING MOUT) is anchored to the vehicle chassis.

The system is defined using a total of 15 points, of which 8 represent the vehicle chassis and are considered fixed. This gives a total of 21 natural coordinates, corresponding to the \(x, y, z\) coordinates of each of the 7 mobile points. Again, this configuration starts from the definition used in the base system description.

A total of 20 constraint equations will be necessary, as there are 21 variables and a single degree of freedom. Of these, eleven equations represent the base of the suspension system, and the remaining 9 define the position of point pushrod outer, located on the upper wishbone, the rod connecting the upper wishbone to the rocker, defined by points pushrod outer and \(pushrod inner\), the rocker defined by points ROCKER PIVOT, ROCKER PIVOT AXIS, and \(Pushrod Inner\), as well as the point l spring mount, which connects to the suspension, and the point U SPRING MOUNT, which connects the other end of the suspension.

Points Defining the Wishbone Suspension Configuration 2

Points Name	Description	Type
wheel center	Center of the Wheel	mobile
UCA FRONT	Upper Control Arm Front	fixed
UCA REAR	Upper Control Arm Rear	fixed
uca outer	Upper Control Arm Outer	mobile
LCA FRONT	Lower Control Arm Front	fixed
LCA REAR	Lower Control Arm Rear	fixed
lca outer	Lower Control Arm Outer	mobile
TIEROD INNER	Inner Tie Rod	fixed
tierod outer	Outer Tie Rod	mobile
pushrod outer	Pushrod Exterior	mobile
pushrod inner	Pushrod Interior	mobile
ROCKER PIVOT	Rocker pivot	fixed
ROCKER PIVOT AXIS	Rocker pivot axis	fixed
U SPRING MOUNT	Spring/damper Upper Mount	fixed
l spring mount	Spring/Damper Lower Mount	mobile

There are a total of 20 constraint equations presented in the table Constraint Equations: Configuration 2 Wishbone Suspension. Again, the first eleven are the same as those used to define the base system.

The equations 1 to 3 define the position of the connection point between the pushrod and the upper wishbone. Equation 4 introduces the condition of constant length of the pushrod. Equations 5 and 6 define the rigid condition of the rocker. It should be noted that the point ROCKER PIVOT is the one that actually connects to the chassis, while ROCKER PIVOT AXIS is an auxiliary point used to define the direction of rotation of the rocker. In other words, a revolute pair has been defined by combining these points. Finally, equations 7 to 9 define the position of point pushrod inner, which connects to the lower end of the suspension.

Constraint Equations: Configuration 2 Wishbone Suspension

Equation	Part Definition	Initial Point	Final Point
1	Upper Wishbone	uca outer	UCA FRONT
2		uca outer	UCA REAR
3	Lower Wishbone	lca outer	LCA FRONT
4		lca outer	LCA REAR
5	Tie Rod	tierod outer	TIEROD INNER
6	Knuckle	tierod outer	uca outer
7		tierod outer	lca outer
8		lca outer	uca outer
9	Wheel Center	wheel center	uca outer
10		wheel center	lca outer
11		wheel center	tierod outer
12	Pushrod point mount	pushrod outer	UCA FRONT
13		pushrod outer	UCA REAR
14		pushrod outer	uca outer
15	Constant distance	pushrod inner	pushrod outer
16	Rocker	pushrod inner	ROCKER PIVOT
17		pushrod inner	ROCKER PIVOT AXIS
18	Suspension Point	l spring mount	UCA FRONT
19		l spring mount	ROCKER PIVOT
20		l spring mount	pushrod inner