This paper presents a novel approach to linearizing the dynamics of Cable-Driven Parallel Robots with a Point-Mass end effector (PM CDPR). The main advantage of this new approach over existing methods is that it greatly simplifies controller design and allows for direct control of the apparent end effector stiffness through adjustment of a single intuitive parameter, making it especially well suited for applications that involve human-robot interaction, such as haptics and rehabilitation. The key element of the linearization process is the inclusion of a constraint on the cable tensions to ensure that the sum of the tension-to-length ratios of all cables remains constant. In addition to outlining the constrained linearization process, this paper also provides a closed-form solution for the workspace boundaries of PM CDPRs with three cables using the proposed constrained dynamic model while respecting cable tension limits.