11.2. Equations of Motion for Undamped MDOF System

Consider an n-DOF linear system with lumped masses and linear springs. The general form of the equations of motion without damping and external forces is: [M]{u¨(t)}+[K]{u(t)}={0}

In this chunk, we introduce an n-degree-of-freedom (n-DOF) linear system, which means that the system can move in n different independent directions. The equation describes how this system behaves when it is either stationary or in motion. The equation itself is comprised of three components:

1. [M]{u¨(t)} refers to the mass matrix multiplied by the acceleration vector. This part describes how the mass of the system resists changes in motion (inertia).
2. [K]{u(t)} refers to the stiffness matrix multiplied by the displacement vector. This part describes how the system reacts when it is displaced from its equilibrium position (elastic force).
3. The equation is set equal to zero, indicating that, in the absence of external forces or damping, the internal forces balance each other out.

You can think of this system like a multi-story building swaying in the wind during a storm. Each floor (or mass) moves independently but is also connected through springs (stiffness). The forces acting on each floor are similar to how the balance of forces keeps everything stable, like a well-designed structure that does not topple over in strong winds.

Where:

• [M] = Mass matrix (n×n)
• [K] = Stiffness matrix (n×n)
• {u(t)} = Displacement vector
• {u¨(t)} = Acceleration vector

Here we define the matrices involved in our equations. The mass matrix [M] captures the distribution of mass across the system. In many practical systems, this matrix is diagonal, meaning each mass primarily affects itself and has minimal interaction with others. The stiffness matrix [K] represents how the structure's components resist deformation. Importantly, it is symmetric and positive-definite, which ensures stable behavior in dynamic analysis. The displacement and acceleration vectors {u(t)} and {u¨(t)} include all the displacements and accelerations of the n masses in our system, respectively.

Imagine a set of building blocks where each block has a different weight (mass) and is connected by rubber bands (stiffness). The way these blocks would move when you pull or compress the rubber bands corresponds to how the mass and stiffness matrices interact in the motion equations.

Key Concepts:

• Mass matrix is usually diagonal in lumped-mass systems.
• Stiffness matrix is symmetric and positive-definite.
• This leads to a system of n coupled second-order differential equations.

In this section, we emphasize the key characteristics of our matrices. A diagonal mass matrix indicates that the forces acting on each mass are primarily independent, simplifying calculations. The symmetry and positive-definiteness of the stiffness matrix ensure that the system's response is well-behaved. The resulting set of equations forms n coupled second-order differential equations, which can be solved to reveal the dynamic behavior of the system over time. Coupling means that changes in one mass's position will affect the others, showcasing the interconnected nature of an MDOF system.

Consider a set of interlinked springs where changing one spring's tension directly impacts the others' tension. The movement of each spring influences its neighbors, much like how the coupled equations represent interactions among different masses in the MDOF system.

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