Multibody Dynamics · Module 1 of 10
Introduction to Multibody Systems and Simulation
A machine is many parts moving together. Multibody dynamics treats it as rigid bodies joined by constraints and pushed by forces, then simulates how the whole thing moves.
Readiness check
Tick only what you can do closed-notes before starting.
- State Newton's second law for a rigid body.
- Recall that a rigid body in a plane has three degrees of freedom.
- Explain what a pin joint does to two connected links.
- Set up a free-body diagram from Statics.
- Recall that a simulation integrates equations of motion in time.
The core idea
A multibody system is rigid bodies connected by joints and loaded by forces. Joints impose constraints that remove degrees of freedom; the motion left over is what the equations of motion predict.
system = bodies + joints + forcesplanar body: 3 freedoms; spatial body: 6DOF = (free coordinates) − (constraint equations)Statics and Dynamics usually study one body at a time. Real machines are many bodies moving together: an engine, a robot arm, a landing gear, a vehicle suspension. Multibody dynamics gives a single, systematic way to describe all of them. Each body carries its own position and orientation. Each joint, a pin, a slider, a ball, ties bodies together and writes down algebraic constraint equations that its connected bodies must satisfy. Forces, gravity, springs, actuators, contact, drive the motion. The structure of the connections is the system topology: an open tree (a robot arm) has no closed loops, while a closed loop (a four-bar linkage) feeds back on itself and needs extra constraints. Counting the free coordinates and subtracting the constraints gives the degrees of freedom, the number of independent motions and the size of the problem the simulator must solve.
The skills, taught in order
Five habits turn a drawing of a machine into a model a computer can simulate.
1.1 Identify the bodies
List every part that moves as a rigid body, and mark the parts that do not move as ground. Each moving planar body brings three coordinates, each spatial body brings six. This count is the raw size of the model before any joint is applied.
1.2 Identify the joints
Every connection is a joint that removes freedoms. A planar pin removes two, a planar slider removes two, a rigid weld removes three. Naming joints correctly is where most modeling errors are caught or created.
1.3 Read the topology
Decide whether the system is an open tree or contains closed loops. Trees are simpler to simulate; closed loops need loop-closure constraints and give rise to the algebraic side of the equations.
1.4 Count degrees of freedom
Subtract the total joint constraints from the total coordinates. The result is the number of inputs you must specify or the number of independent motions the machine has. It also tells you how many initial conditions the simulation needs.
1.5 Frame the simulation
State what drives the system, what you want to predict, and over what time. This turns a static model into a simulation plan: assemble the equations of motion, integrate them in time, then check the answer against physics.
| Element | Adds | Removes |
|---|---|---|
| Planar rigid body | 3 coordinates | none |
| Spatial rigid body | 6 coordinates | none |
| Planar revolute (pin) | none | 2 freedoms |
| Planar prismatic (slider) | none | 2 freedoms |
Bodies add coordinates; joints remove them. The bookkeeping in this table is the whole degree-of-freedom count.
Engineering connection: before any software is opened, this count tells you how many actuators a mechanism needs and whether a design is a mechanism, a structure, or over-constrained.
Worked example 1: degrees of freedom of a four-bar linkage
A planar four-bar linkage has three moving links joined to each other and to ground by four pin joints. How many degrees of freedom does it have, and what does that mean for driving it?
- ProblemFind the mobility of the four-bar linkage in Figure 1.
- Given / findMoving links n = 3 (crank, coupler, rocker), pin joints j = 4. Find the planar degrees of freedom.
- AssumptionsAll links are rigid, all joints are ideal revolute pairs, motion is planar.
- ModelPlanar Grubler count: DOF = 3n − 2j, with n moving bodies and j lower pairs each removing two freedoms.
- EquationsDOF = 3n − 2j
- SolveDOF = 3(3) − 2(4) = 9 − 8 = 1.
- CheckOne degree of freedom means one input angle fully determines the linkage. That matches how a four-bar is driven by a single crank.
- ConclusionThe four-bar is a one-degree-of-freedom mechanism: turn the crank and every other point follows. A single actuator is enough.
Worked example 2: coordinates and constraints of a double pendulum
Model a planar double pendulum as two rigid links pinned end to end, the first pinned to a fixed support. Using absolute coordinates for each body, count coordinates, constraints, and degrees of freedom.
- ProblemFind the degrees of freedom of the double pendulum in Figure 2 using absolute coordinates.
- Given / findTwo moving bodies, each with planar coordinates (x, y, θ). Two pin joints: one to ground, one between the links.
- AssumptionsRigid links, ideal frictionless pins, planar motion.
- ModelCoordinates = 3 per body. Each planar pin removes 2 freedoms. DOF = coordinates − constraints.
- Equationscoordinates = 3 × 2 = 6constraints = 2 × 2 = 4DOF = 6 − 4 = 2
- SolveSix coordinates minus four constraints gives 2 degrees of freedom.
- CheckTwo freedoms match the two joint angles you would use in a minimal model. Absolute coordinates carry extra unknowns plus constraints; both routes agree on the DOF.
- ConclusionThe double pendulum has two degrees of freedom. Absolute coordinates trade a bigger system for a systematic setup, and the constraint count always returns the same physical DOF.
Misconceptions and diagnostics
| Mistake | Symptom | Diagnostic question | Correction |
|---|---|---|---|
| Counting ground as a moving body | DOF comes out too high | Is this part fixed to the world? | Exclude ground from the moving-body count. |
| Forgetting loop-closure constraints | A closed linkage looks like it has too many freedoms | Does the chain form a closed loop? | Add the constraint that closes each loop. |
| Confusing coordinates with DOF | Reporting the number of unknowns as the mobility | Did I subtract the constraints? | DOF = coordinates minus constraint equations. |
| Treating a mechanism as a structure | Expecting no motion from a 1-DOF linkage | Is the DOF zero or positive? | Positive DOF means it moves; zero means it is a structure. |
Practice ladder
A planar mechanism has 5 moving links and 6 revolute joints. Find its degrees of freedom.
Show answer
DOF = 3(5) − 2(6) = 15 − 12 = 3.
A single rigid body floats freely in space with no joints. How many degrees of freedom, and what are they?
Show answer
Six: three translations and three rotations.
A slider is added to the double pendulum tip so it moves along a fixed rail. Using DOF = 2 for the double pendulum, what happens when the rail constraint (removing 1 freedom) is added?
Show answer
DOF becomes 2 − 1 = 1. The rail ties the tip to a line, leaving a single independent motion.
Sketch a machine from your own experience (a bicycle, a desk lamp, a robot gripper). List its bodies, joints, and estimate its degrees of freedom.
What good work looks like
A good answer names each rigid body, classifies each joint by the freedoms it removes, notes any closed loops, and reports DOF = coordinates minus constraints with a sentence on what drives it.
Working with AI, and proving it yourself
Use AI to check your model setup, not to build it
Portfolio task
Take a real machine, list its bodies and joints, count its degrees of freedom, and state what a simulation of it would need as inputs.
Retrieval and spaced review
Closed notes. Answer out loud, then reveal.
1. What three ingredients define a multibody system?
Rigid bodies, joints that constrain them, and forces that drive them.
2. How many freedoms does a planar rigid body have?
Three: two translations and one rotation.
3. Write the planar mobility formula.
DOF = 3n − 2j for n moving bodies and j lower pairs.
4. What distinguishes a tree from a closed-loop system?
A tree has no closed loops; a closed loop needs extra loop-closure constraints.
5. Why count degrees of freedom before simulating?
It sets the number of inputs, initial conditions, and the size of the problem.
Textbook mapping
This course follows Wittenburg, Dynamics of Multibody Systems, using graph concepts to describe system structure. Use these references to read further.
| Topic in this module | Where to read more |
|---|---|
| Multibody systems and system structure | Wittenburg, Dynamics of Multibody Systems, ch. 1 and 5 |
| Systematic body-joint description | Sol, Kinematics and Dynamics of Multibody Systems (1983) |
| Mobility and mechanism analysis | Standard machine-theory texts on the Grubler-Kutzbach criterion |
Chapter references are to Wittenburg, Dynamics of Multibody Systems (Springer). The systematic treatment of arbitrary connections follows Sol (1983).