Short Answer:

In the analysis of a single-degree-of-freedom (SDOF) system, several assumptions are made to simplify the study of vibration behavior. These assumptions help convert a complex real-life system into a simple mathematical model that can be analyzed easily.

The major assumptions are that the system has linear behavior, small displacements, lumped mass, constant stiffness and damping, and motion in only one direction. These assumptions allow engineers to describe the vibration by a single differential equation and make it possible to predict the system’s response accurately for engineering applications.

Detailed Explanation :

Assumptions Made in SDOF Analysis

The single-degree-of-freedom (SDOF) system is a simplified model used in vibration and dynamics studies to represent the motion of a complex system with only one coordinate. In real mechanical or structural systems, vibration can occur in many directions, and components may have distributed mass and stiffness. However, by making a few realistic assumptions, such a system can be idealized into an SDOF model.

These assumptions simplify the mathematical analysis without significantly affecting the accuracy of results for small or moderate vibrations. Below are the main assumptions commonly made in SDOF analysis.

1. The System Has Only One Degree of Freedom

The most basic assumption is that the system’s motion can be completely described using a single independent coordinate such as displacement , velocity , or acceleration .

• This means the system moves or vibrates in only one direction — either translational (linear) or rotational.
• All other motions are either negligible or do not affect the system’s dynamic behavior.
• For example, a spring–mass–damper system moving up and down is a single-degree-of-freedom system.

This assumption helps simplify the mathematical model into a single second-order differential equation representing motion.

2. Linearity of the System

It is assumed that the system behaves linearly, meaning the relationship between force and displacement, velocity, or acceleration follows linear laws.

• The spring obeys Hooke’s law, i.e., , where  is the constant stiffness of the spring.
• The damper produces a viscous damping force proportional to velocity, , where  is the damping coefficient.
• The inertial force follows Newton’s second law, .

Under this assumption, the principle of superposition can be applied — meaning that the system’s total response to multiple forces is the sum of individual responses.

In real systems, materials and damping may exhibit nonlinear behavior at large deformations, but for small vibrations, linearity is a valid and useful assumption.

3. Small Displacements and Small Angles

The displacement of the vibrating body from its equilibrium position is assumed to be very small compared to its overall dimensions.

• Small displacements ensure that restoring forces and motion equations remain linear.
• For systems like pendulums, small angles () are assumed so that the equations remain simple.
• This approximation eliminates higher-order terms in trigonometric or polynomial expressions, reducing the equations to a solvable linear form.

In cases where displacements are large, nonlinear effects such as material stretching, geometric stiffening, or instability must be considered — but these are excluded in SDOF analysis.

4. Mass Is Lumped or Concentrated

In an SDOF system, it is assumed that the entire mass of the system is lumped at a single point or at the center of gravity of the vibrating body.

• The spring and damper are assumed massless and only provide stiffness and damping effects.
• This assumption simplifies the analysis because only one inertia term  is considered.
• In reality, mass may be distributed throughout the system (like in beams or rods), but lumping it at a representative point produces sufficiently accurate results for most engineering calculations.

This lumped-mass assumption transforms a distributed system into a simple, single-particle model, which is easy to analyze.

5. Stiffness Is Constant

The stiffness of the spring or structural element is assumed to remain constant throughout the motion.

• It implies that the restoring force is directly proportional to displacement ().
• This means the spring does not deform plastically, and no permanent deformation occurs.
• Materials are considered to behave elastically, and the stiffness does not change with time or repeated loading.

In real applications, stiffness may vary slightly due to temperature changes, material fatigue, or nonlinear elasticity, but the constant stiffness assumption simplifies the analysis significantly.

6. Damping Is Viscous and Linear

The damping force in the SDOF system is assumed to be viscous and linearly proportional to velocity.

• The damping force is given by , where  is a constant damping coefficient.
• The damping element resists motion and dissipates energy as heat or friction.
• This assumption makes it possible to include damping easily in the equation of motion:

Although real systems may experience other forms of damping (Coulomb friction, structural damping, etc.), viscous damping provides a good approximation for most engineering systems.

7. Material Properties Are Constant

The physical properties of the system — such as mass (m), stiffness (k), and damping (c) — are assumed constant with respect to time and motion.

• This assumption ignores changes due to temperature, wear, or external environmental conditions.
• It ensures that the vibration parameters remain the same throughout the analysis period.

This makes the system time-invariant, meaning its response depends only on the applied force, not on the time-dependent change of properties.

8. No Coupling Between Motions

It is assumed that there is no coupling between translational and rotational motions or between different vibration directions.

• The motion along one axis is independent of motion along any other axis.
• This allows the system to be analyzed purely as one-dimensional.

In multi-degree-of-freedom or continuous systems, coupling effects are important, but they are neglected in the SDOF model for simplicity.

9. Damping Is Small

In most SDOF analyses, damping is assumed to be small compared to stiffness and inertia forces.

• This allows certain simplifications in vibration equations.
• For example, in free vibration, small damping means the natural frequency is approximately equal to the undamped natural frequency.

This assumption makes it possible to find approximate analytical solutions easily.

Conclusion

The analysis of a single-degree-of-freedom system is based on simplifying assumptions such as linearity, small motion, lumped mass, constant stiffness, and viscous damping. These assumptions make it possible to represent complex mechanical or structural systems with a simple mathematical model. Although real-world systems may have nonlinearities or multiple degrees of freedom, these assumptions provide a good starting point for understanding basic vibration behavior and for predicting dynamic responses with acceptable accuracy in engineering design.